geometry assignment find the length of the segment indicated

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Segment Lengths in Circles Fully Explained w/ 10 Examples!

// Last Updated: January 21, 2020 - Watch Video //

Understanding segment lengths in circles is a crucial skill in geometry.

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

You’re in good hands because that’s what today’s lesson is all about.

You’re going to learn how to quickly find the segment lengths (chords, tangents, & secants) for circles.

Let’s get started!

In the previous lesson you learned how to find the arcs given two intersecting secants or tangents.

But now it’s time to investigate the length of two intersecting secants or tangents.

Did you know that three separate theorems help us solve these problems?

It’s true

1. Intersecting Chords Theorem

If two chords or secants intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

As seen in the image below, chords AC and DB intersect inside the circle at point E. Therefore, the product of the lengths of chord AC equals the product of the lengths of chord DB.

Intersecting Chords Theorem

2. Secant Secant Theorem

Now when two secant segments have a common endpoint outside a circle, the product of the measures of one secant segment and its external part is equal to the product of the measures of the other secant and its external part.

As seen in the graphic below, secants GP and FP intersect outside the circle at point P. Therefore, we can solve for unknown lengths by multiplying the external part (PH) by the entire secant length (PG) and set it equal to the product of the external part (PE) and the entire secant length (PF) of the second secant.

Secant Secant Theorem

3. Tangent Secant Theorem

Thirdly, if a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.

The image below illustrates this theorem by showing how the Tangent Secant Theorem and the Secant Secant Theorem are the same: external part (EA) times the whole (EA) is equal to the external part (EC) times the whole (ED).

Tangent Secant Theorem

We could also use the geometric mean to find the length of the secant segment and the length of the tangent segment, as Math Bits Notebook accurately states.

In the video below, you’ll use these three theorems to solve for the length of chords, secants, and tangents of a circle.

Video – Lesson & Examples

• Introduction to Video: Lengths of Intersecting Secants
• 00:00:30 – Theorems for finding segment lengths in circles (Examples #1-4)
• Exclusive Content for Member’s Only
• 00:14:09 – Find the indicated segment length (Examples #5-8)
• 00:28:27 – Find the indicated segment length given secants and tangents (Examples #9-10)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

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Geometry (all content)

Course: geometry (all content)   >   unit 1, measuring line segments.

• Measuring segments
• Congruent line segments
• Congruent segments

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How to Calculate Segment Lengths in Circles Using the Power Theorem

In the vast realm of geometry, circles hold a unique fascination, primarily due to their perfect symmetry and endless periphery. One of the intriguing aspects of circle geometry is the study of segment lengths within these closed curves. Segments, which are areas bound by chords and arcs, vary in length depending on their position and the angles subtended by their bounding chords. The lengths of these segments can reveal a wealth of information about the circle and its associated properties. As we delve deeper into the subject, we'll uncover the mathematical intricacies behind these lengths and explore how they interplay with other geometric elements of the circle, laying the groundwork for further investigations into the more advanced concept of the power theorem.

Step-by-step Guide: Segment Lengths in Circle

1. Chord-Chord Power Theorem : When two chords intersect inside a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord. Mathematically, if \(AB\) and \(CD\) are intersecting chords, then: \( AE \times EB = CE \times ED \)

2. Tangent-Secant Power Theorem : When a tangent and a secant intersect outside a circle, the square of the length of the tangent segment equals the product of the lengths of the secant’s entire length and its external segment. Mathematically, if \(PA\) is a tangent and \(PBC\) is a secant, then: \( PA^2 = PB \times PC \)

3. Secant-Secant Power Theorem : When two secants intersect outside a circle, the product of the lengths of one secant and its external segment equals the product of the lengths of the other secant and its external segment. Mathematically, if \(PAB\) and \(PCD\) are intersecting secants, then: \( PA \times PB = PC \times PD \)

Example 1 : Two intersecting chords in a circle, \(AB\) and \(CD\), are divided into segments of \(3 \text{ cm}\), \(4 \text{ cm}\), \(2 \text{ cm}\), and \(x\) respectively. Using the Chord-Chord Power Theorem, determine the value of \(x\).

Solution : According to the Chord-Chord Power Theorem: \( AE \times EB = CE \times ED \) Given: \( AE = 3 \text{ cm} \) and \( EB = 4 \text{ cm} \) \( 3 \times 4 = 2 \times x \) \( x = 6 \text{ cm} \)

Example 2 : A circle has a tangent \(PA\) of length \(5 \text{ cm}\) and a secant \(PBC\) with segment \(PB\) measuring \(3 \text{ cm}\) and \(PC\) measuring \(7 \text{ cm}\). Validate the Tangent-Secant Power Theorem with the given data.

Solution : According to the Tangent-Secant Power Theorem: \( PA^2 = PB \times PC \) Given: \( PA = 5 \text{ cm} \), \( PB = 3 \text{ cm} \), \( PC = 7 \text{ cm} \) \( 5^2 = 3 \times 7 \) This holds false since \(25 \neq 21\).

Example 3 : Two secants, \(PAB\) and \(PCD\), intersect outside a circle. The given measurements are: \(PA = 2 \text{ cm}\), \(AB = 4 \text{ cm}\), \(PC = 3 \text{ cm}\), and \(CD = y\). Use the Secant-Secant Power Theorem to determine the value of \(y\).

Solution : According to the Secant-Secant Power Theorem: \( PA \times PB = PC \times PD \) Given: \( PB = PA + AB = 2+4 = 6 \text{ cm} \), \( PD = 3 + y \text{ cm} \) \( 2 \times 6 = 3 \times (3+y) \) On solving, \( y = 1 \text{ cm} \).

Practice Questions:

• Two intersecting chords in a circle have lengths of segments as \(2 \text{ cm}\), \(8 \text{ cm}\), \(3 \text{ cm}\) and \(x\). Find \(x\).
• A tangent of length \(6 \text{ cm}\) intersects with a secant of full length \(10 \text{ cm}\) and what is the external segment?
• \( x = 5.33 \text{ cm} \)
• External segment \( = 3.6 \text{ cm} \)

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Length of a Line Segment Calculator

What is a line segment, what is the formula for the length of a line segment, how do i find the length of a line segment with endpoints, how to use the length of a line segment calculator, more distance-related tools.

With the length of a line segment calculator, you can instantly calculate the length of a line segment from its endpoints.

If you're not sure of what a line segment is or how to calculate the length of a segment, then you might like to read the text below. In it, you'll find:

• What is a line segment? ;
• The formula for the length of a line segment ; and
• How to find the length of a segment with its endpoints .

If you glance around, you'll see that we are surrounded by different geometric figures. Perhaps you have a table, a ruler, a pencil, or a piece of paper nearby, all of which can be thought of as geometric figures.

If we look again at the ruler (or imagine one), we can think of it as a rectangle. In geometry, the sides of this rectangle or edges of the ruler are known as line segments. A line segment is one of the basic geometric figures, and it is the main component of all other figures in 2D and 3D.

With these ideas in mind, let's have a look at how the books define a line segment:

"A line segment is a section of a line that has two endpoints, A and B, and a fixed length. Being different from a line, which does not have a beginning or an end. The line segment between points A and B is denoted with a top bar symbol as the segment A B ‾ \overline{AB} A B ."

Returning to the ruler, we could name the beginning of the numbered side as point A and the end as point B. According to the definition, this actually corresponds to a line segment with a beginning and an end (endpoints A and B) and a fixed length (ruler's length).

But what if the line segment we want to calculate the length of isn't the edge of a ruler? Great question! Another way to determine the length of a line segment is by knowing the position (coordinates) of its endpoints A and B.

This implies that a line segment can be drawn in a coordinate plane XY. This coordinate plane representation of a line segment is very useful for algebraically studying the characteristics of geometric figures, as is the case of the length of a line segment.

In the sections below, we go into further detail on how to calculate the length of a segment given the coordinates of its endpoints.

💡 For the sake of convenience, we referred to the endpoints of a line segment as A and B. Endpoints can be labeled with any other letters, such as P and Q, C and F, and so on.

The formula for the length of a line segment is given by the distance formula , an expression derived from the Pythagorean theorem:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

• d — Length of the line segment;
• x₁ and y₁ — Coordinates of any of the endpoints of the line segment; and
• x₂ and y₂ — Coordinates of the other endpoint.

To find the length of a line segment with endpoints:

Use the distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Replace the values for the coordinates of the endpoints, (x₁, y₁) and (x₂, y₂) .

Perform the calculations to get the value of the length of the line segment.

🙋 Not sure if you got the correct result for a problem you're working on? Replace your values in the calculator to verify your answer 😉

With this length of a line segment calculator, you'll be able to instantly find the length of a segment with its endpoints. To use this tool:

In the First point section of the calculator, enter the coordinates of one of the endpoints of the segment, x₁ and y₁ .

Similarly, in the Second point section, input the coordinates' values of the other endpoint, x₂ and y₂ .

Finally, the calculator will display the length of the segment ( Length ) in the Result section.

That's it! 😄

🙋 Why don't you give it a try? What is the length of a line segment with endpoints (-3,1) and (2,5)? 🤔

Did you find the length of a line segment calculator useful? If you did, you might like to visit some of our other distance calculation tools:

Distance calculator ;

2D distance calculator ;

Coordinate distance calculator ;

Euclidean distance calulator ; and

Distance between two points calculator .

What is the length of a line segment from the origin to the point (- 3, 4)?

The length of the line segment is 5 . To obtain this result:

Use the distance formula : d = √[(x₂ - x₁)² + (y₂ - y₁)²]

In our example, the variables of this formula are: (x₁, y₁) = (0, 0) (x₂, y₂) = (-3, 4)

Substitute and perform the corresponding calculations: d = √[(-3 - 0)² + (4 - 0)²] d = √[(-3)² + (4)²] d = √[9 + 16] d = √25

By finding the square root of this number, you get the segment's length: d = 5

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Line Segment

A line segment is a part of a line that has two endpoints and a fixed length. It is different from a line that can be extended in both directions and which does not have a beginning or an end. In this lesson, let us learn more about the line segment definition, the line segment symbol, and how to find the length of a line segment.

What is a Line Segment?

A line segment is a path between two points that can be measured. Since line segments have a defined length, they can form the sides of any polygon. The figure given below shows a line segment AB, where the length of line segment AB refers to the distance between its endpoints, A and B.

Line Segment Definition

A line segment can be defined as a part of a line which a fixed length that can be measured.

Line Segment Symbol

A line segment is represented by a bar on top which is the line segment symbol. It is written as \(\overline{AB}\).

How to Measure Line Segments?

Line segments can be measured with the help of a ruler (scale). Let us see how to measure a given line segment and name it PQ.

• Step 1: Place the tip of the ruler carefully so that zero is placed at the starting point P of the given line segment.
• Step 2: Now, start reading the values given on the ruler and spot the number which comes on the other endpoint Q.
• Step 3: Thus, the length of the line segment is 4 inches, which can be written as \(\overline{PQ}\) = 4 inches.

Line Segment Formula

In the above example, we measured the length of line segment PQ to be 4 inches. This is written as \(\overline{PQ}\) = 4 inches. Now, let us see how to find the length of a line segment when the coordinates of the two endpoints are given. In this case, we use the distance formula , that is, D = √[(\(x_{2}-x_{1}\)) 2 + (\(y_{2}-y_{1}\)) 2 ]. Here, (\(x_{1}\), \(y_{1}\)) and (\(x_{2}\), \(y_{2}\)) are the coordinates of the given points.

For example, a line segment has the following coordinates: (-2, 1) and (4, –3). Let us apply the distance formula to find the length of the line segment. Here, \(x_{1}\) = -2; \(x_{2}\) = 4; \(y_{1}\) = 1; \(y_{2}\) = -3. After substituting these values in the distance formula we get: D =√[(4-(-2)) 2 + (-3-1) 2 ) = √((4+2) 2 + (-3-1) 2 ] = √(6 2 + (-4) 2 ) = √(36 + 16) = √52 = 7.21 units. Therefore, using the distance formula, we found that the length of the line segment with coordinates (-2, 1) and (4, –3) is 7.21 units.

Difference Between Line, Line Segment, and Ray

Observe the figures given below to understand the difference between a line, a line segment, and a ray.

Important Notes

• A line has indefinite ends and cannot be measured.
• A line segment has a start point and an endpoint, thus, it can be measured.
• Line segments have a defined length, hence, they form the sides of any polygon.
• A ray has just one start point and no endpoint, therefore, it cannot be measured.
• The concept of rays can be understood with the example of the rays of the sun, which have a beginning point but no endpoint.

☛Related Topics

Check out the following pages related to the line segment.

• Difference Between Line and Line Segment
• Lines, Rays and Line Segments Worksheets
• How do you know if two line segments are perpendicular?

Line Segment Examples

Example 1: Identify if the given figure is a line segment, a line, or a ray.

The figure has one starting point but an arrow on the other end. This shows that it is not a line segment or a line, it is a ray. Therefore, LM is a ray.

Example 2: Name the line segments in the given triangle.

The line segments which make up the triangle are \(\overline{PQ}\), \(\overline{QR}\), and \(\overline{PR}\). Therefore, the line segments in the given triangle are \(\overline{PQ}\), \(\overline{QR}\), and \(\overline{PR}\).

Example 3: Find the length of the line segment PQ if the coordinates of P and Q are (3, 4) and (2, 0) respectively.

The coordinates of P and Q are (3, 4) and (2, 0). Let us apply the distance formula: D = √[(\(x_{2}-x_{1}\)) 2 + (\(y_{2}-y_{1}\)) 2 ]. Here, \(x_{1}\) = 3; \(x_{2}\) = 2; \(y_{1}\) = 4; \(y_{2}\) = 0. Therefore, the length of the line segment, D =√[(2-3) 2 +(0-4) 2 ] = √((-1) 2 +(-4) 2 ) = √(1 + 16) = √17 = 4.123 units. Therefore \(\overline{PQ}\) = 4.123 units.

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Practice Questions on Line Segment

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FAQs on Line Segment

What is a line segment in math.

A line segment is a part of a line that connects two points which are considered to be its endpoints. It is the distance between two points that can be measured. Since line segments have a defined length, they can form the sides of any polygon .

What is the Difference Between a Line and a Line Segment?

A line has no endpoints and can be extended in both ends, whereas, a line segment has two fixed endpoints. A line is infinite, whereas, a line segment has a defined length and so, it can be measured.

How to Draw a Line Segment?

There are mainly two methods to construct a line segment of any length. The first method of constructing a line segment is simple in which we take a ruler (scale) and mark the starting point as P, then we need to mark the endpoint as Q, with the required length of the line segment. For example, if the required length is 4 inches, we mark Q at 4 inches with the help of the ruler. After this step, the two points are joined together which shows the line segment of the desired length. Visit the Methods to Draw a Line Segment page for a detailed explanation.

What is the Midpoint of a Line Segment?

The midpoint of a line segment refers to a point that divides it into two equal parts and is located in the middle of the line segment.

How to Find the Midpoint of a Line Segment?

The midpoint of a line segment can be calculated if the coordinates of the endpoints are given. For example, if (x 1 , y 1 ) and (y 1, y 2 ) are the two endpoints, then the midpoint of the line segment can be calculated by the formula, Midpoint = [(x 1 + x 2 )/2, (y 1 + y 2 )/2]

What are the Examples of Line Segments in Real Life?

We know that line segments have a fixed length or measure. Thus, the examples of line segments in real life include the sides of a polygon, the edges of a ruler, the edges of a paper, etc.

What is the Symbol of a Line Segment?

A line segment is denoted by a bar on top (—) like \(\overline{AB}\). This bar is considered as the line segment symbol.

How to Find the Length of a Line Segment?

In order to find the length of a line segment, we use a scale (ruler) to measure its dimensions. In some cases, if the coordinates of the endpoints of the line segment are given, then we apply the distance formula , D = √[(\(x_{2}-x_{1}\)) 2 + (\(y_{2}-y_{1}\)) 2 ], where 'D' is the distance between the endpoints of the line segment and (\(x_{1}\), \(y_{1}\)) and (\(x_{2}\), \(y_{2}\)) are the coordinates of the two points.

When are Line Segments Congruent?

Two figures are said to be congruent if they are of the same size and shape. Thus, any two line segments can be considered to be congruent if they are of the same length.

Can a Line Segment be Extended?

No, a line segment cannot be extended because it has a fixed and definite length. Although a line can be extended in two opposite directions indefinitely, and a ray can also be extended from one end.

What does a Line Segment Look Like?

A line segment looks like a straight line with fixed endpoints. For example, if we observe the sides of any polygon, like a triangle , or a square , or even a pentagon , we can see that the sides of these polygons are line segments.

• May 30, 2020

How to find length of a line segment

Updated: Jun 22, 2021

Dear Secondary Math students, Math Lobby will be teaching you how to find length of a line segment on a graph (or just with the coordinates of its end points). By mathematics definition in layman terms, a line segment is part of a line with two distinct end points. Let’s begin!

In this note, you will learn:

· how to find the length of a line segment given the coordinates of its end points.

Given that we have a random line segment AB on a graph,

To begin, we need to find the coordinates of the end-points of the line segment if we are not provided with it. From the graph above, we can clearly see that the coordinates of the end-points A and B to be (2,1) and (8,4) respectively.

So now, how can we use this piece of information to find the length of the line segment? We might need a little help from a chapter we have previously covered: Pythagoras’ Theorem ! (If you have not viewed our article on the Pythagoras’ Theorem, you can do so here!

How can the concept of Pythagoras’ Theorem aid us in solving this question in this case? Take a look at the end-points of the line, if we were to extrapolate the y-coordinate of A and x-coordinate of B, does it form something familiar?

Yes! You are right, it is a right-angled triangle! With this new-found information, we can apply the concept of Pythagoras’ Theorem to solve for the length of the line segment. Let’s see how it’s done:

Through extrapolation, we can see that the y-coordinates of A and the x-coordinate of B intersects at the point (8,1) to give us a right-angled triangle. Hence, let’s name this intersection C. Now, let’s recap the formula we learnt back in Pythagoras’ Theorem: a² + b² = c² , which in this case, a and b are essentially just AC and BC, and AB is essentially just the hypotenuse of the right-angled triangle!

How do we find the length of AC and BC then? Simple! To get the length of AC and BC, just simply subtract the x-coordinate of A by the x-coordinate of C and the y-coordinates of C by the y-coordinate of B respectively! This will give us a newly-derived formula that resembles the formula of Pythagoras’ Theorem, which we call it the distance formula .

Distance Formula = √ [ (x1 – x2)² + (y1 – y2)² ]

(pythagoras’ theorem solving for the hypotenuse, c = √( a2 + b2 )).

*Note: The order of subtraction (ie. whether to use point A to subtract point C or point C to subtract point A) does not matter, because 1) you will end up with the same values just with the exception of one being negative, 2) the squaring of the values will take out any negative signs and 3) we will always take the positive value because we know that length can never be negative!

Length of AB = √ [(8 -2)² + (4-1)²]

= 6.7082039325 units

~ 6.71 units (corrected to 3 s.f)

And that’s how you find the length of a line segment with the coordinates of the end-points!

Now, let’s try some examples:

1) Given that we have a line segment DE, where D (5, 12) and E (8, 10). Find the length of the line segment.

Answer: Length of DE = 3.61 units (corrected to 3 s.f) or √13 units

2) Given that we have a line segment FG, where F (-3, 15) and G (9, -4). Find the length of the line segment.

Answer: Length of FG = 22.5 units (corrected to 3 s.f) or √505 units

And that’s all we have for today, math students! Be sure to check out our other math articles as well on Math Lobby, and we hope that you understood the explanation and correct method to find the length of a line segment in a graph with the coordinates of the end-points.

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Angle Properties of Circles

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2022 N Level Math Syllabus Objectives

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• Review of equations
• Simplifying square roots
• Adding and subtracting square roots
• Multiplying square roots
• Dividing square roots
• Line segments and their measures inches
• Line segments and their measures cm
• Segment Addition Postulate
• Angles and their measures
• Classifying angles
• Naming angles
• The Angle Addition Postulate
• Angle pair relationships
• Understanding geometric diagrams and notation
• Parallel lines and transversals
• Proving lines parallel
• Points in the coordinate plane
• The Midpoint Formula
• The Distance Formula
• Parallel lines in the coordinate plane
• Classifying triangles
• Triangle angle sum
• The Exterior Angle Theorem
• Triangles and congruence
• SSS and SAS congruence
• ASA and AAS congruence
• SSS, SAS, ASA, and AAS congruences combined
• Right triangle congruence
• Isosceles and equilateral triangles
• Midsegment of a triangle
• Angle bisectors
• The Triangle Inequality Theorem
• Inequalities in one triangle
• Classifying quadrilaterals
• Angles in quadrilaterals
• Properties of parallelograms
• Properties of trapezoids
• Properties of rhombuses
• Properties of kites
• Areas of triangles and quadrilaterals
• Introduction to polygons
• Polygons and angles
• Areas of regular polygons
• Solving proportions
• Similar polygons
• Using similar polygons
• Similar triangles
• Similar right triangles
• Proportional parts in triangles and parallel lines
• The Pythagorean Theorem and its Converse
• Multi-step Pythagorean Theorem problems
• Special right triangles
• Multi-step special right triangle problems
• Trig. ratios
• Inverse trig. ratios
• Solving right triangles
• Multi-step trig. problems
• Rhombuses and kites with right triangles
• Trigonometry and area
• Identifying solid figures
• Volume of prisms and cylinders
• Surface area of prisms and cylinders
• Volume of pyramids and cones
• Surface area of pyramids and cones
• More on nets of solids
• Similar solids
• Arcs and central angles
• Arcs and chords
• Circumference and area
• Inscribed angles
• Tangents to circles
• Secant angles
• Secant-tangent and tangent-tangent angles
• Segment measures
• Equations of circles
• Translations
• Reflections
• All transformations combined
• Sample spaces and The Counting Principle
• Independent and dependent events
• Mutualy exclusive events
• Permutations
• Combinations
• Permutations vs combinations
• Probability using permutations and combinations
• Line segments
• Perpendicular segments
• Medians of triangles
• Altitudes of triangles