problem solving pythagorean theorem

Module 11: Geometry

Using the pythagorean theorem to solve problems, learning outcomes.

• Use the pythagorean theorem to find the unknown length of a right triangle given the two other lengths

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around [latex]500[/latex] BCE.

Remember that a right triangle has a [latex]90^\circ [/latex] angle, which we usually mark with a small square in the corner. The side of the triangle opposite the [latex]90^\circ [/latex] angle is called the hypotenuse, and the other two sides are called the legs. See the triangles below.

In a right triangle, the side opposite the [latex]90^\circ [/latex] angle is called the hypotenuse and each of the other sides is called a leg.

Study Guides > Prealgebra

Using the pythagorean theorem to solve problems, learning outcomes.

• Use the pythagorean theorem to find the unknown length of a right triangle given the two other lengths

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around [latex]500[/latex] BCE. Remember that a right triangle has a [latex]90^\circ [/latex] angle, which we usually mark with a small square in the corner. The side of the triangle opposite the [latex]90^\circ [/latex] angle is called the hypotenuse, and the other two sides are called the legs. See the triangles below.

The Pythagorean Theorem

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MathBootCamps

The pythagorean theorem with examples.

The Pythagorean theorem is a way of relating the leg lengths of a right triangle to the length of the hypotenuse, which is the side opposite the right angle. Even though it is written in these terms, it can be used to find any of the side as long as you know the lengths of the other two sides. In this lesson, we will look at several different types of examples of applying this theorem.

Table of Contents

• Examples of using the Pythagorean theorem
• Solving applied problems (word problems)
• Solving algebraic problems

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Applying the Pythagorean theorem (examples)

In the examples below, we will see how to apply this rule to find any side of a right triangle triangle. As in the formula below, we will let a and b be the lengths of the legs and c be the length of the hypotenuse. Remember though, that you could use any variables to represent these lengths.

In each example, pay close attention to the information given and what we are trying to find. This helps you determine the correct values to use in the different parts of the formula.

Find the value of \(x\).

The side opposite the right angle is the side labelled \(x\). This is the hypotenuse. When applying the Pythagorean theorem, this squared is equal to the sum of the other two sides squared. Mathematically, this means:

\(6^2 + 8^2 = x^2\)

Which is the same as:

\(100 = x^2\)

Therefore, we can write:

\(\begin{align}x &= \sqrt{100}\\ &= \bbox[border: 1px solid black; padding: 2px]{10}\end{align}\)

Maybe you remember that in an equation like this, \(x\) could also be –10, since –10 squared is also 100. But, the length of any side of a triangle can never be negative and therefore we only consider the positive square root.

In other situations, you will be trying to find the length of one of the legs of a right triangle. You can still use the Pythagorean theorem in these types of problems, but you will need to be careful about the order you use the values in the formula.

Find the value of \(y\).

The side opposite the right angle has a length of 12. Therefore, we will write:

\(8^2 + y^2 = 12^2\)

This is the same as:

\(64 + y^2 = 144\)

Subtracting 64 from both sides:

\(y^2 = 80\)

\(\begin{align}y &= \sqrt{80} \\ &= \sqrt{16 \times 5} \\ &= \bbox[border: 1px solid black; padding: 2px]{4\sqrt{5}}\end{align}\)

In this last example, we left the answer in exact form instead of finding a decimal approximation. This is common unless you are working on an applied problem.

Applications (word problems) with the Pythagorean theorem

There are many different kinds of real-life problems that can be solved using the Pythagorean theorem. The easiest way to see that you should be applying this theorem is by drawing a picture of whatever situation is described.

Two hikers leave a cabin at the same time, one heading due south and the other headed due west. After one hour, the hiker walking south has covered 2.8 miles and the hiker walking west has covered 3.1 miles. At that moment, what is the shortest distance between the two hikers?

First, sketch a picture of the information given. Label any unknown value with a variable name, like x.

Due south and due west form a right angle, and the shortest distance between any two points is a straight line. Therefore, we can apply the Pythagorean theorem and write:

\(3.1^2 + 2.8^2 = x^2\)

Here, you will need to use a calculator to simplify the left-hand side:

\(17.45 = x^2\)

Now use your calculator to take the square root. You will likely need to round your answer.

\(\begin{align}x &= \sqrt{17.45} \\ &\approx 4.18 \text{ miles}\end{align}\)

As you can see, it will be up to you to determine that a right angle is part of the situation given in the word problem. If it isn’t, then you can’t use the Pythagorean theorem.

Algebra style problems with the Pythagorean theorem

There is one last type of problem you might run into where you use the Pythagorean theorem to write some type of algebraic expression. This is something that you will not need to do in every course, but it does come up.

A right triangle has a hypotenuse of length \(2x\), a leg of length \(x\), and a leg of length y. Write an expression that shows the value of \(y\) in terms of \(x\).

Since no figure was given, your first step should be to draw one. The order of the legs isn’t important, but remember that the hypotenuse is opposite the right angle.

Now you can apply the Pythagorean theorem to write:

\(x^2 + y^2 = (2x)^2\)

Squaring the right-hand side:

\(x^2 + y^2 = 4x^2\)

When the problem says “the value of \(y\)”, it means you must solve for \(y\). Therefore, we will write:

\(y^2 = 4x^2 – x^2\)

Combining like terms:

\(y^2 = 3x^2\)

Now, use the square root to write:

\(y = \sqrt{3x^2}\)

Finally, this simplifies to give us the expression we are looking for:

\(y = \bbox[border: 1px solid black; padding: 2px]{x\sqrt{3x}}\)

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The Pythagorean theorem allows you to find the length of any of the three sides of a right triangle. It is one of those things that you should memorize, as it comes up in all areas of math, and therefore in many different math courses you will probably take. Remember to avoid the common mistake of mixing up where the legs go in the formula vs. the hypotenuse and to always draw a picture when one isn’t given.

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Word problems on Pythagorean Theorem

Learn how to solve different types of word problems on Pythagorean Theorem .

Pythagoras Theorem can be used to solve the problems step-by-step when we know the length of two sides of a right angled triangle and we need to get the length of the third side.

Three cases of word problems on Pythagorean Theorem :

Case 1: To find the hypotenuse where perpendicular and base are given.

Case 2: To find the base where perpendicular and hypotenuse are given.

Case 3: To find the perpendicular where base and hypotenuse are given.

Word problems using the Pythagorean Theorem:

1. A person has to walk 100 m to go from position X in the north of east direction to the position B and then to the west of Y to reach finally at position Z. The position Z is situated at the north of X and at a distance of 60 m from X. Find the distance between X and Y.

⇒ 200x = 10000 + 3600

⇒ 200x = 13600

⇒ x = 13600/200

Therefore, distance between X and Y = 68 meters.

Therefore, length of each side is 8 cm.

Using the formula solve more word problems on Pythagorean Theorem.

3. Find the perimeter of a rectangle whose length is 150 m and the diagonal is 170 m.

In a rectangle, each angle measures 90°.

Therefore PSR is right angled at S

Using Pythagoras theorem, we get

⇒ PS = √6400

Therefore perimeter of the rectangle PQRS = 2 (length + width)

                                                          = 2 (150 + 80) m

                                                          = 2 (230) m

                                                          = 460 m

4. A ladder 13 m long is placed on the ground in such a way that it touches the top of a vertical wall 12 m high. Find the distance of the foot of the ladder from the bottom of the wall.

Let the required distance be x meters. Here, the ladder, the wall and the ground from a right-angled triangle. The ladder is the hypotenuse of that triangle.

According to Pythagorean Theorem,

Therefore, distance of the foot of the ladder from the bottom of the wall = 5 meters.

5. The height of two building is 34 m and 29 m respectively. If the distance between the two building is 12 m, find the distance between their tops.

The vertical buildings AB and CD are 34 m and 29 m respectively.

Draw DE ┴ AB

Then AE = AB – EB but EB = BC

Therefore AE = 34 m - 29 m = 5 m

Now, AED is right angled triangle and right angled at E.

⇒ AD = √169

Therefore the distance between their tops = 13 m.

The examples will help us to solve various types of word problems on Pythagorean Theorem.

Congruent Shapes

Congruent Line-segments

Congruent Angles

Congruent Triangles

Conditions for the Congruence of Triangles

Side Side Side Congruence

Side Angle Side Congruence

Angle Side Angle Congruence

Angle Angle Side Congruence

Right Angle Hypotenuse Side congruence

Pythagorean Theorem

Proof of Pythagorean Theorem

Converse of Pythagorean Theorem

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Pythagorean Theorem

How to Use The Pythagorean Theorem

The Formula

The picture below shows the formula for the Pythagorean theorem. For the purposes of the formula, side $$ \overline{c}$$ is always the hypotenuse . Remember that this formula only applies to right triangles .

Examples of the Pythagorean Theorem

When you use the Pythagorean theorem, just remember that the hypotenuse is always 'C' in the formula above. Look at the following examples to see pictures of the formula.

Conceptual Animation of Pythagorean Theorem

Demonstration #1.

More on the Pythagorean theorem

Demonstration #2

Video tutorial on how to use the pythagorean theorem.

Step By Step Examples of Using the Pythagorean Theorem

Example 1 (solving for the hypotenuse).

Use the Pythagorean theorem to determine the length of X.

Identify the legs and the hypotenuse of the right triangle .

The legs have length 6 and 8 . $$X $$ is the hypotenuse because it is opposite the right angle.

Substitute values into the formula (remember 'C' is the hypotenuse).

$ A^2+ B^2= \red C^2 \\ 6^2+ 8^2= \red X^2 $

$A^2+ B^2= \red X^2 \\ 100= \red X^2 \\ \sqrt {100} = \red X \\ 10= \red X $

Example 2 (solving for a Leg)

The legs have length 24 and $$X$$ are the legs. The hypotenuse is 26.

$ \red A^2+ B^2= C^2 \\ \red x^2 + 24^2= {26}^2 $

$ \red x^2 + 24^2= 26^2 \\ \red x^2 + 576= 676 \\ \red x^2 = 676 - 576 \\ \red x^2 = 100 \\ \red x = \sqrt { 100} \\ \red x = 10 $

Practice Problems

Find the length of X.

Remember our steps for how to use this theorem. This problems is like example 1 because we are solving for the hypotenuse .

The legs have length 14 and 48 . The hypotenuse is X.

$ A^2 + B^2 = C^2 \\ 14^2 + 48^2 = x^2 $

Solve for the unknown.

$ 14^2 + 48^2 = x^2 \\ 196 + 2304 = x^2 \\ \sqrt{2500} = x \\ \boxed{ 50 = x} $

Use the Pythagorean theorem to calculate the value of X. Round your answer to the nearest tenth.

Remember our steps for how to use this theorem. This problems is like example 2 because we are solving for one of the legs .

The legs have length 9 and X . The hypotenuse is 10.

$ A^2 + B^2 = C^2 \\ 9^2 + x^2 = 10^2 $

$ 9^2 + x^2 = 10^2 \\ 81 + x^2 = 100 \\ x^2 = 100 - 81 \\ x^2 = 19 \\ x = \sqrt{19} \approx 4.4 $

Use the Pythagorean theorem to calculate the value of X. Round your answer to the nearest hundredth.

The legs have length '10' and 'X'. The hypotenuse is 20.

$ A^2 + B^2 = C^2 \\ 10^2 + \red x^2 = 20^2 $

$ 10^2 + \red x^2 = 20^2 \\ 100 + \red x^2 = 400 \\ \red x^2 = 400 -100 \\ \red x^2 = 300 \\ \red x = \sqrt{300} \approx 17.32 $

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Pythagorean Theorem and Problems with Solutions

Explore some simple proofs of the Pythagorean theorem and its converse and use them to solve problems. Detailed solutions to the problems are also presented.

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Pythagorean Theorem – Definition, Formula, Problems

In mathematic, the Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of its other two sides . Another way of stating the theorem is that the sum of the areas of the squares formed by the sides of a right triangle equals the area of the square whose side is the hypotenuse. The theorem is a key relation in Euclidean geometry. It is named for the Greek philosopher Pythagorus.

Remember: The Pythagorean theorem only applies to right triangles!

Pythagorean Theorem Formula

The formula for the Pythagorean theorem describes the relationship between the sides a and b of a right triangle to its hypotenuse, c . A right triangle is one containing a 90° or right angle. The hypotenuse is the side of the triangle opposite from the right angle (which is the largest angle in a right triangle).

a 2 + b 2 = c 2

Solving for a, b, and c

Rearranging the equation gives the formulas solving for a, b, and c:

• a = (c 2 – b 2 ) ½
• b = (c 2 – a 2 ) ½
• c = (a 2 + b 2 ) ½

How to Solve the Pythagorean Theorem – Example Problems

For example, find the hypotenuse of a right triangle with side that have lengths of 5 and 12.

Start with the formula for the Pythagorean theorem and plug in the numbers for the sides a and b to solve for c .

a 2 + b 2 = c 2 5 2 + 12 2 = c 2 c 2 = 5 2 + 12 2 = 25 + 144 = 169 c2 = 169 c = √169 or 169 ½ = 13

For example, solve for side b of a triangle where a is 9 and the hypotenuse c is 15.

a 2 + b 2 = c 2 9 2 + b 2 = 15 2 b 2 = 15 2 – 9 2 = 225 – 81 = 144 b = √144 = 12

Now, let’s combine a bit of algebra with the geometry. Solve for x where the sides of a right triangle are 5x and 4x +5 and the hypotenuse has a length of 8x -3.

a 2 + b 2 = c 2 (5x) 2 + (4x +5) 2 = (8x-3) 2

The (4x + 5) 2 and (8x -3)2 terms are the squares of binomial expressions. So, expanding the equation gives the following:

25x 2 + (4x +5)(4x +5) = (8x -3)(8x -3) 25x 2 = 16×2 + 20x +20x + 25 = 64x – 24x – 24x + 9

Combine like terms:

41x 2 + 40x + 25 = 64x 2 – 48x + 9

Rewrite the equation and solve for x.

0 = 23x 2 – 88x – 16

Apply the quadratic equation and solve for x:

x = [-b ± √(b 2 -4ac)]/2a x = [-(-88) ± √[-88 2 – 4(23)(-16)] / 2(23) = [88 ± √(7744 + 1472)] / 46 = (88 ± 96) / 46

So, there are two answers:

x = (88 + 96)/46 = 4 and (88 – 96).46 = -4/23

A triangle does not have a negative length for its side, so x is 4.

Plugging in”4″ in place of x, the sides of the right triangle are 20, 21, and 29.

Pythagorean Triples

Pythagorean triples are integers a, b, and c, that represent the sides of a right triangle and satisfy the Pythagorean theorem. Here is the list of Pythagorean triples for integers with values less than 100:

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

Proof of the Pythagorean Theorem

There are more proofs for the Pythagorean theorem than for any other theorem in geometry! At least 370 proofs are known. Some of these proofs use the parallel postulate. Some rely on the complementarity of acute angles in a right triangle. Proofs using shearing use the properties of parallelograms.

History – Did Pythagoras Discover the Pythagorean Theorem?

While the Pythagorean theorem takes its name from Pythagorus, he did not discover it. Exactly who gets the credit or whether many different places made the discovery independently is a matter of debate. The Mesopotamians made calculations using the formula as early as 2000 BC, which was over a thousand years before Pythagorus. A papyrus from the Egyptian Middle Kingdom, dating between 2000 and 1786 BC, references a math problem describing Pythagorean triples. The Baudhayana Shulba Sutra from India (dating between the 8th and 5th century BC) lists both Pythagorean triples and the Pythagorean theorem. The “Gougu theorem” from China offers a proof for the Pythagorean theorem, which came into use long before its oldest surviving description from the 1st century BC.

Pythagorus of Samos lived between 570 and 495 BC. While he was not the original person who formulated the Pythagorean theorem, he (or his students) may have introduced its proof to ancient Greece. In any case, his philosophical treatment of math left a lasting impression on the world.

• Bell, John L. (1999). The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development . Kluwer. ISBN 0-7923-5972-0.
• Heath, Sir Thomas (1921). “ The ‘Theorem of Pythagoras ‘”. A History of Greek Mathematics (2 Vols.) (Dover Publications, Inc. (1981) ed.). Oxford: Clarendon Press. ISBN 0-486-24073-8.
• Maor, Eli (2007). The Pythagorean Theorem: A 4,000-Year History . Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-12526-8.
• Swetz, Frank; Kao, T. I. (1977). Was Pythagoras Chinese?: An Examination of Right Triangle Theory in Ancient China . Pennsylvania State University Press. ISBN 0-271-01238-2.

Related Posts

Pythagoras Theorem

The Pythagoras theorem which is also referred to as the Pythagorean theorem explains the relationship between the three sides of a right-angled triangle. According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides of a triangle. Let us learn more about the Pythagoras theorem, the Pythagoras theorem formula , and the proof of Pythagoras theorem along with examples.

What is the Pythagoras Theorem?

The Pythagoras theorem states that if a triangle is a right-angled triangle, then the square of the hypotenuse is equal to the sum of the squares of the other two sides. Observe the following triangle ABC, in which we have BC 2 = AB 2 + AC 2 ​​. Here, ​​​​AB is the base, AC is the altitude (height), and BC is the hypotenuse. It is to be noted that the hypotenuse is the longest side of a right-angled triangle.

Pythagoras Theorem Equation

The Pythagoras theorem equation is expressed as, c 2 = a 2 + b 2 , where 'c' = hypotenuse of the right triangle and 'a' and 'b' are the other two legs. Hence, any triangle with one angle equal to 90 degrees produces a Pythagoras triangle and the Pythagoras equation can be applied in the triangle.

History of Pythagoras Theorem

Pythagoras theorem was introduced by the Greek Mathematician Pythagoras of Samos. He was an ancient Greek philosopher who formed a group of mathematicians who worked religiously on numbers and lived like monks. Although Pythagoras introduced the theorem, there is evidence that proves that it existed in other civilizations too, 1000 years before Pythagoras was born. The oldest known evidence is seen between the 20th to the 16th century B.C in the Old Babylonian Period.

Pythagoras Theorem Formula

The Pythagorean theorem formula states that in a right triangle ABC, the square of the hypotenuse is equal to the sum of the squares of the other two legs. If AB and AC are the sides and BC is the hypotenuse of the triangle, then: BC 2 = AB 2 + AC 2 ​. In this case, AB is the base, AC is the altitude or the height, and BC is the hypotenuse.

Another way to understand the Pythagorean theorem formula is using the following figure which shows that the area of the square formed by the longest side of the right triangle (the hypotenuse) is equal to the sum of the area of the squares formed by the other two sides of the right triangle.

In a right-angled triangle, the Pythagoras Theorem Formula is expressed as:

c 2 = a 2 + b 2

• 'c' = hypotenuse of the right triangle
• 'a' and 'b' are the other two legs.

Pythagoras Theorem Proof

The Pythagoras theorem can be proved in many ways. Some of the most common and widely used methods are the algebraic method and the similar triangles method. Let us have a look at both these methods individually in order to understand the proof of this theorem.

Proof of Pythagorean Theorem Formula using the Algebraic Method

The proof of the Pythagoras theorem can be derived using the algebraic method. For example, let us use the values a, b, and c as shown in the following figure and follow the steps given below:

• Step 1: This method is also known as the 'proof by rearrangement'. Take 4 congruent right-angled triangles, with side lengths 'a' and 'b', and hypotenuse length 'c'. Arrange them in such a way that the hypotenuses of all the triangles form a tilted square. It can be seen that in the square PQRS, the length of the sides is 'a + b'. The four right triangles have 'b' as the base, 'a' as the height and, 'c' as the hypotenuse.
• Step 2: The 4 triangles form the inner square WXYZ as shown, with 'c' as the four sides.
• Step 3: The area of the square WXYZ by arranging the four triangles is c 2 .
• Step 4: The area of the square PQRS with side (a + b) = Area of 4 triangles + Area of the square WXYZ with side 'c'. This means (a + b) 2 = [4 × 1/2 × (a × b)] + c 2 .This leads to a 2 + b 2 + 2ab = 2ab + c 2 . Therefore, a 2 + b 2 = c 2 . Hence, the Pythagoras theorem formula is proved.

Pythagorean Theorem Formula Proof using Similar Triangles

Two triangles are said to be similar if their corresponding angles are of equal measure and their corresponding sides are in the same ratio. Also, if the angles are of the same measure, then by using the sine law, we can say that the corresponding sides will also be in the same ratio. Hence, corresponding angles in similar triangles lead us to equal ratios of side lengths.

Derivation of Pythagorean Theorem Formula

Consider a right-angled triangle ABC, right-angled at B. Draw a perpendicular BD meeting AC at D.

In △ABD and △ACB,

• ∠A = ∠A (common)
• ∠ADB = ∠ABC (both are right angles)

Thus, △ABD ∼ △ACB (by AA similarity criterion)

Similarly, we can prove △BCD ∼ △ACB.

Thus △ABD ∼ △ACB, Therefore, AD/AB = AB/AC. We can say that AD × AC = AB 2 .

Similarly, △BCD ∼ △ACB. Therefore, CD/BC = BC/AC. We can also say that CD × AC = BC 2 .

Adding these 2 equations, we get AB 2 + BC 2 = (AD × AC) + (CD × AC)

AB 2 + BC 2 =AC(AD +DC)

AB 2 + BC 2 =AC 2

Hence proved.

Pythagoras Theorem Triangles

Right triangles follow the rule of the Pythagoras theorem and they are called Pythagoras theorem triangles. The three sides of such a triangle are collectively called Pythagoras triples . All the Pythagoras theorem triangles follow the Pythagoras theorem which says that the square of the hypotenuse is equal to the sum of the two sides of the right-angled triangle. This can be expressed as c 2 = a 2 + b 2 ; where 'c' is the hypotenuse and 'a' and 'b' are the two legs of the triangle.

Pythagoras Theorem Squares

As per the Pythagorean theorem, the area of the square which is built upon the hypotenuse of a right triangle is equal to the sum of the area of the squares built upon the other two sides. These squares are known as Pythagoras squares.

Applications of Pythagoras Theorem

The applications of the Pythagoras theorem can be seen in our day-to-day life. Here are some of the applications of the Pythagoras theorem.

• Engineering and Construction fields

Most architects use the technique of the Pythagorean theorem to find the unknown dimensions. When the length or breadth is known it is very easy to calculate the diameter of a particular sector. It is mainly used in two dimensions in engineering fields.

• Face recognition in security cameras

The face recognition feature in security cameras uses the concept of the Pythagorean theorem, that is, the distance between the security camera and the location of the person is noted and well-projected through the lens using the concept.

• Woodwork and interior designing

The Pythagoras concept is applied in interior designing and the architecture of houses and buildings.

People traveling in the sea use this technique to find the shortest distance and route to proceed to their concerned places.

• Right Triangle Formulas
• Hypotenuse Leg Theorem
• Similar Triangles
• Pythagoras Theorem Worksheets

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Pythagorean Theorem Examples

Example 1: The hypotenuse of a right-angled triangle is 16 units and one of the sides of the triangle is 8 units. Find the measure of the third side using the Pythagoras theorem formula.

Given: Hypotenuse = 16 units

Let us consider the given side of a triangle as the perpendicular height = 8 units

On substituting the given dimensions to the Pythagoras theorem formula

Hypotenuse 2 = Base 2 + Height 2

16 2 = B 2 + 8 2

B 2 = 256 - 64

B = √192 = 13.856 units

Therefore, the measure of the third side of the triangle is 13.856 units.

Example 2: Julie wanted to wash her building window which is 12 feet off the ground. She has a ladder that is 13 feet long. How far should she place the base of the ladder away from the building?

We can visualize this scenario as a right triangle. We need to find the base of the right triangle formed. We know that, Hypotenuse 2 = Base 2 + Height 2 . Thus, we can say that b 2 = 13 2 - 12 2 where 'b' is the distance of the base of the ladder from the feet of the wall of the building. So, b 2 = 13 2 - 12 2 can be solved as, b 2 = 169 - 144 = 25. This means, b = √25 = 5. Hence, we get 'b' = 5.

Therefore, the base of the ladder is 5 feet away from the building.

Example 3: Use the Pythagoras theorem to find the hypotenuse of the triangle in which the sides are 8 units and 6 units respectively.

Using the Pythagoras theorem, Hypotenuse 2 = Base 2 + Height 2 = 8 2 + 6 2 . This leads to Hypotenuse 2 = 64 + 36 = 100. Therefore, hypotenuse = √100 = 10 units.

Therefore, the length of the hypotenuse is 10 units.

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Practice Questions on Pythagoras Theorem

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FAQs on Pythagoras Theorem

What is the pythagoras theorem in math.

The Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem can be expressed as, c 2 = a 2 + b 2 ; where 'c' is the hypotenuse and 'a' and 'b' are the two legs of the triangle. These triangles are also known as Pythagoras theorem triangles.

What is the Converse of Pythagoras Theorem?

The converse of Pythagoras theorem is: If the sum of the squares of any two sides of a triangle is equal to the square to the third (largest) side, then it is said to be a right-angled triangle .

What is the Use of the Pythagorean Theorem Formula?

The Pythagoras theorem works only for right-angled triangles. When any two values are known, we can apply the Pythagoras theorem and calculate the unknown sides of the triangle. There are other real-life applications of the Pythagoras theorem like in the field of navigation, engineering and architecture.

What is the use of the Pythagoras Theorem?

The Pythagorean theorem is used in various fields. A few of its applications are given below.

• Architecture, construction and Navigation industries.
• For computing the distance between points on the plane.
• For calculating the perimeter, the surface area, the volume of geometrical shapes, and so on.

Can the Pythagorean Theorem Formula be Applied to any Triangle?

No, the Pythagorean theorem can only be applied to a right-angled triangle since the Pythagorean theorem expresses the relationship between the sides of the triangle where the square of the two legs is equal to the square of the third side which is the hypotenuse.

How to Work Out the Pythagoras Theorem?

Pythagoras theorem can be used to find the unknown side of a right-angled triangle. For example, if two legs of a right-angled triangle are given as 4 units and 6 units, then the hypotenuse (the third side) can be calculated using the formula, c 2 = a 2 + b 2 ; where 'c' is the hypotenuse and 'a' and 'b' are the two legs. Substituting the values in the formula, c 2 = a 2 + b 2 = c 2 = 4 2 + 6 2 = 16 + 36 = √52 = 7.2 units.

What is the Formula of Pythagoras Theorem?

The formula of Pythagoras theorem is expressed as, Hypotenuse 2 = Base 2 + Height 2 . This is also written as, c 2 = a 2 + b 2 ; where 'c' is the hypotenuse and 'a' and 'b' are the two legs of the right-angled triangle . Using the Pythagoras theorem formula, any unknown side of a right-angled can be calculated if the other two sides are given.

Why is the Pythagoras Theorem Important?

The Pythagoras theorem is important because it helps in calculating the unknown side of a right-angled triangle. It has other real-life applications in the field of architecture and engineering, navigation, and so on.

How is Pythagoras Theorem used in Navigation?

The Pythagoras theorem is commonly used in air navigation and ship navigation. The Pythagoras theorem provides a way to the ship's navigator to calculate the distance to a point in the ocean, for example, if the distance between two points is given as 600 km north and 800 km west, the required distance can be calculated using the Pythagoras theorem.

When is Pythagoras Theorem used?

Pythagoras theorem is used when any two sides of a right-angled triangle are given and the third side needs to be calculated. For example, if the perpendicular and base of a right-angled triangle are given as 12 units and 5 units respectively, and we need to find the third side (the hypotenuse) we can calculate it using the theorem which says hypotenuse 2 = perpendicular 2 + base 2 . After substituting the values in the equation we get hypotenuse 2 = 12 2 + 5 2 = 144 + 25 = 169. So, hypotenuse = √169 = 13 units.

What is the Pythagoras Property of Triangles?

The Pythagoras property of triangles is another term for the Pythagoras theorem. According to the Pythagoras property, in a right-angled triangle, the square of the hypotenuse is always equal to the sum of the squares of the other two sides. This theorem is expressed as, c 2 = a 2 + b 2 ; where 'c' is the hypotenuse and 'a' and 'b' are the two legs of the triangle.

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How to Solve Pythagorean Theorem Word Problems

The Pythagorean Theorem is a cornerstone of mathematics, fundamental in various scientific fields and real-life situations. We will guide you through understanding and solving Pythagorean Theorem word problems, explaining each step meticulously for your convenience.

The Pythagorean Theorem is an equation attributed to the ancient Greek mathematician, Pythagoras. The theorem states that in any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The formula can be written as \(a^2+b^2=c^2\).

An essential part of understanding the Pythagorean Theorem involves recognizing Pythagorean triplets. These are sets of three integers that satisfy the Pythagorean Theorem. Notable examples are \((3,4,5), (5,12,13)\), and \((8,15,17)\). Recognizing these sets can simplify solving Pythagorean Theorem word problems.

A Step-by-step Guide to Solving Pythagorean Theorem Word Problems

Now that we’ve laid the groundwork, let’s delve into the process of solving word problems involving the Pythagorean Theorem.

Step 1: Analyzing the Problem

The first step in solving any word problem is understanding what the problem is asking. Translate the problem into an understandable format and identify the elements related to the Pythagorean Theorem.

Step 2: Sketching the Problem

Visualizing the problem can be extremely helpful. Sketch the triangle described in the problem and label the sides according to the given information. The right angle is usually denoted by a small square, and the hypotenuse, by the letter c .

Step 3: Applying the Pythagorean Theorem

Apply the Pythagorean Theorem to the problem. Plug in the values of the known sides into the theorem and solve for the unknown side.

Step 4: Verifying the Solution

After you’ve found the value of the unknown side, verify your solution by substituting the values back into the theorem. If the sides satisfy the theorem, you’ve correctly solved the problem.

Example of a Pythagorean Theorem Word Problem

Consider a problem where you’re given a right-angled triangle with one side measuring \(5\) units and the hypotenuse measuring \(13\) units. You’re asked to find the length of the other side.

You can use the Pythagorean Theorem to solve this problem. By substituting \(a=5\) and \(c=13\) into the theorem, you can solve for \(b\).

Calculating this will give you \(b^2=13^2-5^2=144\). Therefore, \(b=\sqrt{144}=12\) units. Your verification will involve substituting \(a=5, b=12\), and \(c=13\) into the theorem. The equation \(5^2+12^2=13^2\) checks out, confirming the solution.

The Pythagorean Theorem is a potent tool for solving geometrical problems involving right-angled triangles. With the steps outlined in this guide, you can confidently tackle any word problem that comes your way. Practice is key to mastery, so take time to solve different problems and apply the theorem in real-world situations.

by: Effortless Math Team about 10 months ago (category: Articles )

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Pythagoras Theorem

Pythagoras Theorem (also called Pythagorean Theorem) is an important topic in Mathematics, which explains the relation between the sides of a right-angled triangle . The sides of the right triangle are also called Pythagorean triples. The formula and proof of this theorem are explained here with examples.

Pythagoras theorem is basically used to find the length of an unknown side and the angle of a triangle. By this theorem, we can derive the base, perpendicular and hypotenuse formulas. Let us learn the mathematics of the Pythagorean theorem in detail here.

Pythagoras Theorem Statement

Pythagoras theorem states that “ In a right-angled triangle,  the square of the hypotenuse side is equal to the sum of squares of the other two sides “. The sides of this triangle have been named Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say a, b and c) which have positive integer values, when squared, are put into an equation, also called a Pythagorean triple.

The theorem is named after a Greek Mathematician called Pythagoras.

Pythagoras Theorem Formula

Consider the triangle given above:

Where “a” is the perpendicular,

“b” is the base,

“c” is the hypotenuse.

According to the definition, the Pythagoras Theorem formula is given as:

The side opposite to the right angle (90°)  is the longest side (known as Hypotenuse) because the side opposite to the greatest angle is the longest.

Consider three squares of sides a, b, c mounted on the three sides of a triangle having the same sides as shown.

By Pythagoras Theorem –

Area of square “a” + Area of square “b” = Area of square “c”

The examples of theorem and based on the statement given for right triangles is given below:

Consider a right triangle, given below:

Find the value of x.

X is the side opposite to the right angle, hence it is a hypotenuse.

Now, by the theorem we know;

Hypotenuse 2  = Base 2 + Perpendicular 2

x 2 = 8 2 + 6 2

x 2 = 64+36 = 100

x = √100 = 10

Therefore, the value of x is 10.

Pythagoras Theorem Proof

Given: A right-angled triangle ABC, right-angled at B.

To Prove- AC 2 = AB 2 + BC 2

Construction: Draw a perpendicular BD meeting AC at D.

We know, △ ADB ~ △ABC

Therefore, \(\begin{array}{l}\frac{AD}{AB}=\frac{AB}{AC}\end{array} \) (corresponding sides of similar triangles)

Or, AB 2  = AD × AC …………………………….. ……..(1)

Also, △ BDC ~△ABC

Therefore, \(\begin{array}{l}\frac{CD}{BC}=\frac{BC}{AC}\end{array} \) (corresponding sides of similar triangles)

Or, BC 2 = CD × AC ……………………………… ……..(2)

Adding the equations (1) and (2) we get,

AB 2  + BC 2  = AD × AC + CD × AC

AB 2  + BC 2  = AC (AD + CD)

Since, AD + CD = AC

Therefore, AC 2 = AB 2 + BC 2

Hence, the Pythagorean theorem is proved.

Note:   Pythagorean theorem is only applicable to Right-Angled triangle.

Video Lesson on Pythagoras Theorem

Applications of Pythagoras Theorem

• To know if the triangle is a right-angled triangle or not.
• In a right-angled triangle, we can calculate the length of any side if the other two sides are given.
• To find the diagonal of a square.

Pythagoras theorem is useful to find the sides of a right-angled triangle. If we know the two sides of a right triangle, then we can find the third side.

How to use Pythagoras Theorem?

To use Pythagoras theorem, remember the formula given below:

c 2 = a 2  + b 2

Where a, b and c are the sides of the right triangle.

For example, if the sides of a triangles are a, b and c, such that a = 3 cm, b = 4 cm and c is the hypotenuse. Find the value of c.

c 2 = 3 2 +4 2

c 2  = 9+16

Hence, the length of hypotenuse is 5 cm.

How to find whether a triangle is a right-angled triangle?

If we are provided with the length of three sides of a triangle, then to find whether the triangle is a right-angled triangle or not, we need to use the Pythagorean theorem.

Let us understand this statement with the help of an example.

Suppose a triangle with sides 10cm, 24cm, and 26cm are given. 

Clearly, 26 is the longest side.

It also satisfies the condition, 10 + 24 > 26

c 2  = a 2  + b 2     ………(1)

So, let a = 10, b = 24 and c = 26

First we will solve R.H.S. of equation 1.

a 2   + b 2 = 10 2  + 24 2  = 100 + 576 = 676

Now, taking L.H.S, we get;

c 2 = 26 2  = 676

We can see, 

Therefore, the given triangle is a right triangle, as it satisfies the Pythagoras theorem.

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Pythagorean theorem solved examples.

Problem 1:  The sides of a triangle are 5, 12 & 13 units. Check if it has a right angle or not.

Solution: From Pythagoras Theorem, we have;

Perpendicular 2 + Base 2 = Hypotenuse 2

P 2 + B 2 = H 2

Perpendicular (P) = 12 units

Base (B)= 5 units

Hypotenuse (H) = 13 units {since it is the longest side measure}

LHS = P 2 + B 2

⇒ 12 2 + 5 2

⇒ 169 = 169 

L.H.S. = R.H.S.

Therefore, the angle opposite to the 13 units side will be a right angle.

Problem 2:  The two sides of a right-angled triangle are given as shown in the figure. Find the third side.

Perpendicular = 15 cm

Base = b cm

Hypotenuse = 17 cm

As per the Pythagorean Theorem, we have;

⇒15 2 + b 2 = 17 2

⇒225 + b 2 = 289

⇒b 2 = 289 – 225

Therefore,  b = 8 cm

Problem 3:  Given the side of a square to be 4 cm. Find the length of the diagonal .

Solution-  Given;

Sides of a square = 4 cm

To Find- The length of diagonal ac.

Consider triangle abc (or can also be acd)

(ab) 2 +(bc) 2  = (ac) 2

(4) 2 +(4) 2 = (ac) 2

16 + 16 = (ac) 2

32 = (ac) 2

(ac) 2 = 32

Thus, the length of the diagonal is  4√2 cm.

Practice Problems on Pythagoras Theorem

• In a right triangle ABC, right-angled at B, the lengths of AB and BC are 7 units and 24 units, respectively. Find AC.
• If the length of the diagonal of a square is 10 cm, then find the length of the side of the square.
• A triangle is given whose sides are of length 11 cm, 60 cm, and 61 cm. Check whether these are the sides of a right-angled triangle.

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Frequently Asked Questions on Pythagoras Theorem

What is the formula for pythagorean theorem.

The formula for Pythagoras, for a right-angled triangle, is given by; P 2 + B 2 = H 2

What does Pythagoras theorem state?

Pythagoras theorem states that, in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides.

What is the formula for hypotenuse?

The hypotenuse is the longest side of the right-angled triangle, opposite to right angle, which is adjacent to base and perpendicular. Let base, perpendicular and hypotenuse be a, b and c respectively. Then the hypotenuse formula, from the Pythagoras statement will be; c  = √(a 2  + b 2 )

Can we apply the Pythagoras Theorem for any triangle?

No, this theorem is applicable only for the right-angled triangle.

What is the use of Pythagoras theorem?

Can the diagonals of a square be found using pythagoras theorem.

Yes, the diagonals of a square can be found using the Pythagoras theorem, as the diagonal divides the square into right triangles.

Explain the steps involved in finding the sides of a right triangle using Pythagoras theorem.

Step 1: To find the unknown sides of a right triangle, plug the known values in the Pythagoras theorem formula. Step 2: Simplify the equation to find the unknown side. Step 3: Solve the equation for the unknown side.

What are the different ways to prove Pythagoras theorem?

There are various approaches to prove the Pythagoras theorem. A few of them are listed below: Proof using similar triangles Proof using differentials Euclid’s proof Algebraic proof, and so on.

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Pythagoras' Theorem

Over 2000 years ago there was an amazing discovery about triangles:

When a triangle has a right angle (90°) ...

... and squares are made on each of the three sides, ...

... then the biggest square has the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation:

a 2 + b 2 = c 2

• c is the longest side of the triangle
• a and b are the other two sides

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let's see if it really works using an example.

Example: A "3, 4, 5" triangle has a right angle in it.

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side . (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:

Then we use algebra to find any missing value, as in these examples:

Example: Solve this triangle

Read Builder's Mathematics to see practical uses for this.

Also read about Squares and Square Roots to find out why √ 169 = 13

Example: Solve this triangle.

Example: what is the diagonal distance across a square of size 1.

It works the other way around, too: when the three sides of a triangle make a 2 + b 2 = c 2 , then the triangle is right angled.

Example: Does this triangle have a Right Angle?

Does a 2 + b 2 = c 2 ?

• a 2 + b 2 = 10 2 + 24 2 = 100 + 576 = 676
• c 2 = 26 2 = 676

They are equal, so ...

Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle?

Does 8 2 + 15 2 = 16 2 ?

• 8 2 + 15 2 = 64 + 225 = 289 ,
• but 16 2 = 256

So, NO, it does not have a Right Angle

Yes, it does!

So this is a right-angled triangle

And You Can Prove The Theorem Yourself !

Get paper pen and scissors, then using the following animation as a guide:

• Draw a right angled triangle on the paper, leaving plenty of space.
• Draw a square along the hypotenuse (the longest side)
• Draw the same sized square on the other side of the hypotenuse
• Draw lines as shown on the animation, like this:

• Cut out the shapes
• Arrange them so that you can prove that the big square has the same area as the two squares on the other sides

Another, Amazingly Simple, Proof

Here is one of the oldest proofs that the square on the long side has the same area as the other squares.

Watch the animation, and pay attention when the triangles start sliding around.

You may want to watch the animation a few times to understand what is happening.

The purple triangle is the important one.

We also have a proof by adding up the areas .

Pythagorean Theorem Calculator

How to use the pythagorean theorem calculator, what is the pythagorean theorem, how do i use pythagorean theorem, what is the hypotenuse formula, other considerations when dealing with triangles, the pythagorean theorem calculator in the making.

This Pythagorean theorem calculator will calculate the length of any of the missing sides of a right triangle, provided you know the lengths of its other two sides. This includes calculating the hypotenuse. The hypotenuse of the right triangle is the side opposite the right angle, and is the longest side. This side can be found using the hypotenuse formula, another term for the Pythagorean theorem when it's solving for the hypotenuse.

Recall that a right triangle is a triangle with an angle measuring 90 degrees. The other two angles must also total 90 degrees, as the sum of the measures of the angles of any triangle is 180. Read on to answer "what is the Pythagorean theorem and how is it used?"

The Pythagorean theorem calculator is one of the most accessible tools you will come across, despite the name being scary. All you need is any two of the three sides of a right triangle, and you are all set.

Let's take a look at the steps to use our Pythagorean theorem calculator.

Input leg a of the right triangle.

Next, input leg b of the right triangle.

And that is it. The tool will display the following results:

a. The hypotenuse (c) of the right triangle.

b. The area of the triangle.

c. And the perimeter of the triangle.

You don't necessarily have to input legs a and b. If you know the hypotenuse value and need to know any other legs, input the information accordingly.

The default unit is centimeter (cm) for length and cm² for area. You can change these to one of the listed of options according to your requirements. Remember to change the unit before you input the values.

How about we try to understand an example ? For instance, you are climbing a ladder to your roof, and you get curious about the length of the ladder . You notice that the distance from the roof to the ground is leg a , and the distance from the wall to the ladder's base is leg b . And the ladder itself is the hypotenuse. You input the measurements in the tool as leg a and b, 4 m and 3 m , respectively. (Remember to change the units from centimeters to meters first). The calculator instantly determines the hypotenuse as 5 m , the area as 6 m² , and the perimeter as 12 m .

Next time you get curious about any setup that looks like a right triangle, you can use our Pythagorean theorem calculator to determine the lengths of the legs.

The Pythagorean theorem describes how the three sides of a right triangle are related in Euclidean geometry. It states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse . You can also think of this theorem as the hypotenuse formula. If the legs of a right triangle are a and b and the hypotenuse is c , the formula is:

a² + b² = c²

The theorem was credited to the ancient Greek philosopher and mathematician Pythagoras, who lived in the sixth century BC. Although it was previously used by the Indians and Babylonians, Pythagoras (or his students) were credited with being the first to prove the theorem. It should be noted that there is no concrete evidence that Pythagoras himself worked on or proved this theorem.

Here's how to use Pythagorean theorem:

• Input the two lengths that you have into the formula. For example, suppose you know one leg a = 4 and the hypotenuse c = 8.94 . We want to find the length of the other leg b .
• After the values are put into the formula, we have 4² + b² = 8.94² .
• Square each term to get 16 + b² = 80 .
• Combine like terms to get b² = 64 .
• Take the square root of both sides of the equation to get b = 8 . Go ahead and check it with an online Pythagorean theorem calculator!

Note that if you are solving for a or b , rearrange the equation to isolate the desired variable before combining like terms and taking the square root

The Pythagorean theorem calculator will solve for the sides in the same manner that we displayed above. We have included the method to show you how you can solve your problem if you prefer to do it by hand.

The hypotenuse formula simply takes the Pythagorean theorem and solves for the hypotenuse, c . To solve for the hypotenuse, we simply take the square root of both sides of the equation a² + b² = c² and solve for c . When doing so, we get c = √(a² + b²) . This is just a reformulation of the Pythagorean theorem and is often associated with the name hypotenuse formula .

Notice the sides of a triangle have a certain degree of gradient or slope. We can use the slope calculator to determine the slope of each side. In a right triangle, the sides that form the right angle will have slopes whose product is -1. The formula for slope, if you wish to calculate by hand, is:

(y₂ − y₁)/(x₂ − x₁)

You can also figure out the missing side lengths and angles of a right triangle using the right triangle calculator . If the angles given in the problem are in degrees and you want to convert to radians or radians to degrees, check out our angle converter . There is an easy way to convert degrees to radians and radians to degrees.

If the angle is in radians:

• Multiply by 180/π .

If the angle is in degrees:

• Multiply by π/180 .

Sometimes you may encounter a problem where two lengths are missing. In such cases, the Pythagorean theorem calculator won't help – you will use trigonometric functions to solve for these missing pieces. Don't worry! We have an excellent trigonometric functions calculator available for you.

Indeed, all maths enthusiasts would be happy to have access to a Pythagorean theorem calculator. Even the students who have to complete their assignments would be thrilled. Now imagine how happy Mateusz and Piotr were when they decided to make a tool for one of the most sought-after mathematics concepts and successfully did so.

Mateusz Mucha is the brain behind Omni Calculator. His deep love for numbers with strategic vision and operational expertise is a testament to his career. He believes in a hands-on approach in all aspects of life, whether it is leadership, building some innovative calculator or digital product, or participating in a cycling marathon. He is a well-balanced blend of exemplary leadership and vision, with strategic thinking, innovation, and attention to detail being a few of the skills in his arsenal.

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What is the hypotenuse given legs 7 and 9?

The hypotenuse is 11.40 .

You need to apply the Pythagorean theorem:

• Recall the formula a² + b² = c² , where a , and b are the legs and c is the hypotenuse.
• Put the length of the legs into the formula: 7² + 9² = c² .
• Squaring gives 49 + 81 = c² . That is, c² = 150 .
• Taking the square root, we obtain c = 11.40 . You can verify the result with an online Pythagorean theorem calculator.

What is the leg in an isosceles triangle with hypotenuse 10?

Each leg has length 10/√2 ≈ 7.07 . To arrive at this answer, we apply the Pythagorean theorem:

• In our case, a = b , so the formula reads 2a² = c² .
• Solving for a , we get a = c/√2 .
• Plugging in c = 10 , we get the final answer: a = 10/√2 ≈ 7.07 .

30 60 90 triangle

Black hole collision, center of mass, ideal egg boiling.

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