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1.1: Pythagorean Theorem
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Discover, geometrically prove, and apply the Pythagorean Theorem.
Lengths of Triangle Sides Using the Pythagorean Theorem
You've just signed up to be an architect's assistant in a new office downtown. You're asked to draw a scale model of a sculpture for a business plaza. The sculpture has a large triangular piece where one of the angles between the sides is ninety degrees. This type of triangle is called a ‘‘right triangle’’. The architect you're working for comes into the room and tells you that the sides of the triangle that form the right angle are 9 feet and 12 feet. Can you tell how long the third side is?
Finding the Length of Triangle Sides Using Pythagorean Theorem
From Geometry, recall that the Pythagorean Theorem is \(a^2+b^2=c^2\) where \(a\) and \(b\) are the legs of a right triangle and \(c\) is the hypotenuse. Also, the side opposite the angle is lower case and the angle is upper case. For example, angle \(A\) is opposite side \(a\).
The Pythagorean Theorem is used to solve for the sides of a right triangle.
Using the Pythagorean Theorem
\(a=8\), \(b=15\), we need to find the hypotenuse.
\(\begin{aligned} 8^2+15^2&=c^2 \\ 64+225&=c^2 \\ 289&=c^2 \\ 17&=c \end{aligned}\)
Notice, we do not include -17 as a solution because a negative number cannot be a side of a triangle.
Use the Pythagorean Theorem to find the missing leg.
\(\begin{aligned} (5\sqrt{7})^2+x^2&=(5\sqrt{13})^2 \\ 25\cdot 7+x^2&=25\cdot 13 \\ 175+x^2&=325 \\ x^2&=150 \\ x&=5\sqrt{6}\end{aligned}\)
3. Use the Pythagorean Theorem to find the missing leg in the triangle above.
\(\begin{aligned} 10^2+x^2&=(10\sqrt{2})^2 \\ 100+x^2&=100\cdot 2 \\ 100+x^2&=100 \\ x^2&=100 \\ x&=10 \end{aligned}\)
Example \(\PageIndex{1}\)
Earlier, you were given a problem asking you to draw a scale model of a sculpture for a business plaza.
With your knowledge of the Pythagorean Theorem, you can see that the triangle has sides with lengths 9 feet and 12 feet. You work to find the hypotenuse:
\(\begin{aligned} a^2+b^2&=c^2 \\ 9^2+12^2&=c^2 \\ 81+144&=c^2 \\ 225&=c^2 \\ (\sqrt{225})&=15=c \end{aligned}\)
With the knowledge that the length of the third side of the triangle is 15 feet, you are able to construct your scale model with ease.
Example \(\PageIndex{2}\)
Use the Pythagorean Theorem to find the missing side of the following triangle:
\(a=1\), \(b=8\), we need to find the hypotenuse.
\(\begin{aligned} 1^2+8^2&=c^2 \\ 1+64&=c^2 \\ 65&=c^2 \\ \sqrt{65}&=c\end{aligned}\)
Example \(\PageIndex{3}\)
\(a=3\), \(b=11\), we need to find the length of side c, the hypotenuse.
\(\begin{aligned} 3^2+11^2&=c^2 \\ 9+121&=c^2 \\ 130&=c^2 \\ \sqrt{130}&=c\end{aligned}\)
Example \(\PageIndex{4}\)
Find the missing side of the right triangle below. Leave the answer in simplest radical form.
\(a=7\), \(c=18\), we need to find the length of side b.
\(\begin{aligned}7^2+b^2=18^2 \\ 49+b^2=18^2 \\ 324−49=b^2 \\ 275=b^2 \\ \sqrt{275}=b \end{aligned}\)
Find the missing sides of the right triangles. Leave answers in simplest radical form.
- If the legs of a right triangle are 3 and 4, then the hypotenuse is _____________.
- If the legs of a right triangle are 6 and 8, then the hypotenuse is _____________.
- If the legs of a right triangle are 5 and 12, then the hypotenuse is _____________.
- If the sides of a square are length 6, then the diagonal is _____________.
- If the sides of a square are 9, then the diagonal is _____________.
- If the sides of a square are \(x\), then the diagonal is _____________.
- If the legs of a right triangle are 3 and 7, then the hypotenuse is _____________.
- If the legs of a right triangle are \(2\sqrt{5}\) and 6, then the hypotenuse is _____________.
- If one leg of a right triangle is 4 and the hypotenuse is 8, then the other leg is _____________.
- If one leg of a right triangle is 10 and the hypotenuse is 15, then the other leg is _____________.
- If one leg of a right triangle is \(4\sqrt{7}\) and the hypotenuse is \(10\sqrt{6}\), then the other leg is _____________.
- If the legs of a right triangle are x and y, then the hypotenuse is ____________.
Pythagorean Theorem Proof
Use the picture below to answer the following questions.
- Find the area of the square in the picture with sides (a+b).
- Find the sum of the areas of the square with sides c and the right triangles with legs a and b.
- Explain why the areas found in the previous two problems should be the same value. Then, set the expressions equal to each other and simplify to get the Pythagorean Theorem.
Review (Answers)
To see the Review answers, open this PDF file and look for section 1.1.
Additional Resources
Interactive element.
Video: Proving the Pythagorean Theorem
Practice: Pythagorean Theorem
Trigonometry (Algebra 2 Curriculum - Unit 12) | All Things Algebra®
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Due to the length of this Trigonometry Unit Bundle , it is divided into two parts with two unit tests. In addition to the unit tests, each part includes guided notes, homework assignments, quizzes, and study guides to cover the following topics:
Unit 12 Part I:
• Pythagorean Theorem
• Special Right Triangles
• Trigonometric Functions (sin, cos, tan, csc, sec, cot)
• Finding Side and Angle Measures
• Applications: Angle of Elevation and Depression
• Angles in Standard Position
• Converting between Degrees and Radians
• Coterminal and Reference Angles
• Trigonometric Functions in the Coordinate Plane
• The Unit Circle
• Law of Sines
• Law of Cosines
• Area of Triangles
• Applications of Law of Sines, Law of Cosines, and Area
Unit 12 Part II:
• Graphing Trigonometric Functions
• Trigonometric Identities
• Sum and Difference of Angle Identities
• Double-Angle and Half-Angle Identities
• Solving Trigonometric Equations
ADDITIONAL COMPONENTS INCLUDED:
(1) Links to Instructional Videos: Links to videos of each lesson in the unit are included. Videos were created by fellow teachers for their students using the guided notes and shared in March 2020 when schools closed with no notice. Please watch through first before sharing with your students. Many teachers still use these in emergency substitute situations. (2) Editable Assessments: Editable versions of each quiz and the unit test are included. PowerPoint is required to edit these files. Individual problems can be changed to create multiple versions of the assessment. The layout of the assessment itself is not editable. If your Equation Editor is incompatible with mine (I use MathType), simply delete my equation and insert your own.
(3) Google Slides Version of the PDF: The second page of the Video links document contains a link to a Google Slides version of the PDF. Each page is set to the background in Google Slides. There are no text boxes; this is the PDF in Google Slides. I am unable to do text boxes at this time but hope this saves you a step if you wish to use it in Slides instead!
This resource is included in the following bundle(s):
Algebra 2 Curriculum
More Algebra 2 Units:
Unit 1 – Equations and Inequalities
Unit 2 – Linear Functions and Systems
Unit 3 – Parent Functions and Transformations
Unit 4 – Solving Quadratics and Complex Numbers
Unit 5 – Polynomial Functions
Unit 6 – Radical Functions
Unit 7 – Exponential and Logarithmic Functions
Unit 8 – Rational Functions
Unit 9 – Conic Sections
Unit 10 – Sequences and Series
Unit 11 – Probability and Statistics
LICENSING TERMS: This purchase includes a license for one teacher only for personal use in their classroom. Licenses are non-transferable , meaning they can not be passed from one teacher to another. No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses. If you are a coach, principal, or district interested in transferable licenses to accommodate yearly staff changes, please contact me for a quote at [email protected].
COPYRIGHT TERMS: This resource may not be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.
© All Things Algebra (Gina Wilson), 2012-present
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Name: Unit 12: Trigonometry Date: Bell: Homework 1: Pythagorean Theorem, Special Right Triangles, & Trig Functions ** This is a 2-page document ** Directions: Find each missing length. Give all answers in simplest radical form. 1. 16 14 18 10 3. 4. 2.10 14,5 5. 30 60 28 7. Find the values of the six trigonometric functions for a 8.
The Pythagorean theorem is a mathematical formula that describes the relationship between the sides of a right triangle. It is used to find the length of a missing side or to check if a triangle is a right triangle.. The theorem states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In trigonometry, special right triangles are ...
Find an answer to your question Can anyone answer this Unit 8:Right Triangles&Trigonometry Homework 1 Pythagorean theorem and its converse See what teachers have to say about Brainly's new learning tools! ... has anyone done Unit 12 Trigonometry Homework 1 Pythagorean Theorem 2 page homework? heart. 1. Does anyone have the answers for this ...
in a right triangle, the side that makeup the right angle. Pythagorean Theorem. in a right triangle, the sum of the squares of the two legs is equal to the squares of the hypotenuse. Hypotenuse. longest side of a right triangle, always opposite the right angle. The equation for the Pythagorean theorem is a + b = c.
Unit 8 Part 1 - Pythagorean Triples, Pythagorean Theorem and its Converse, Special Right Triangles ... gerometry b unit 4 lesson 1 the pythagorean theorem and its converse. 5 terms. dingerbugal1993. Preview. Geometry Vocab. 47 terms. Elizabeth_Moore3. Preview. chapter 8 quadrilaterals. 13 terms. quizlette48008477. Preview. Trigonometric Ratios ...
TOPIC HOMEWORK DAY 1 Pythagorean Theorem, Special Right Triangles, ... Angles in Standard Position, Converting Degrees and Radians, Coterminal Angles, Reference Angles HW #3 DAY 4 The Unit Circle HW #4 DAY 5 Quiz 12-1 None DAY 6 Law of Sines ... DAY 19 Solving Trigonometric Equations HW #14 DAY 20 Unit 12 Review (Part II) Study for ...
Using the Pythagorean Theorem. 1. Figure 1.1.2 1.1. 2. a = 8 a = 8, b = 15 b = 15, we need to find the hypotenuse. 82 +152 64 + 225 289 17 = c2 = c2 = c2 = c 8 2 + 15 2 = c 2 64 + 225 = c 2 289 = c 2 17 = c. Notice, we do not include -17 as a solution because a negative number cannot be a side of a triangle. 2.
Here's the Pythagorean Theorem formula for your quick reference. Note: drawings not to scale. Problem 1: Find the value of [latex]x [/latex] in the right triangle. Problem 2: Find the value of [latex]x [/latex] in the right triangle. Problem 3: Find the value of [latex]x [/latex] in the right triangle. Problem 4: The legs of a right triangle ...
The lengths of the first two triangles in the trigonometry questions are as follows: 17.2 and 8.2 respectively. How to find the missing lengths. According to the Pythagorean theorem, c² = a² + b². Now for the first triangle where the adjacent is 10 and the opposite 14, we would arrive at the hypotenuse this way: c² = 10² + 14²
Section 7.1. Pythagorean Theorem and Its Converse. G.2.3 Use the triangle angle sum theorem and/or the Pythagorean Theorem and its converse, to solve simple triangle problems. and justify results;
In addition to the unit tests, each part includes guided notes, homework assignments, quizzes, and study guides to cover the following topics: Unit 12 Part I: • Pythagorean Theorem. • Special Right Triangles. • Trigonometric Functions (sin, cos, tan, csc, sec, cot) • Finding Side and Angle Measures. • Applications: Angle of Elevation ...
11 The Pythagorean Theorem Key Concepts Theorem 8-1 Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a2 +b2 =c2 a b c 1. 32 ±42 ≠52 2. 52 ±122 ≠132 62 ±82 ≠102 42 ±42 ≠(4 )"2 2 Check Skills You'll Need GO for Help Vocabulary Tip ...
find the length of the missing leg of a right triangle given a leg of length 8 a hypotenuse of length 10. leave your answer in simplest radical form. 6. does the set of numbers 13, 21, and 24 form a Pythagorean triple? explain. no; 13^2+21^2=/24^2. a triangle has side lengths of 12 cm, 15cm, and 20cm. classify it as acute, obtuse or right. obtuse.
Leave your answer in simplest radical form. 6. Does the set of numbers 13, 21, and 24 form a Pythagorean triple? Explain. no; 13^2+21^2≠24^2. A triangle has side lengths of 12 cm, 15 cm, and 20 cm. Classify it as acute, obtuse, or right. obtuse. A gardener wants to divide a square piece of lawn in half diagonally.
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In the case of a right triangle, a²+b²=c². Converse of the Pythagorean Theorem. If the angles are summative in terms of a²+b²=c², it is a right triangle. Pythagorean Triple. Three integers that, as side lengths of a triangle, form a right triangle (Ex. 3/4/5 or 5/12/13) 3-4-5. Pythagorean Triple.
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Right Triangle. c^2 = a^2 + b^2. angle of elevation. angle formed by a horizontal line and a line of sight to a point above the line. angle of depression. angle formed by a horizontal line and a line of sight to a point below the line. Study with Quizlet and memorize flashcards containing terms like Pythagorean Theorem, Converse of the ...
Geometry questions and answers; Name: Date: Unit 8: Right Triangles & Trigonometry Per: Homework 1: Pythagorean Theorem and its Converse This is a 2-page document Directions: Find the value of x. 1. 2. I 19 10 . 21 7 3 . 4. 16 12.8 27 5.3 5. 6. 20 19 18 31 7. 44 16 22 8. Scott is using a 12-foot ramp to help load furniture into the back of a ...