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Explore Exploring Segment Length Relationships in Circles

Any segment connecting two points on a circle is a chord. In some cases, two chords drawn inside the same circle will intersect, creating four segments. In the following activity, you will look for a pattern in how these segments are related and form a conjecture.

A Using geometry software or a compass and straightedge, construct circle A with two chords

_ CD and

_ EF that intersect inside the circle. Label the intersection point G.

B Repeat your construction with two more circles. Vary the size of the circles and where you put the intersecting chords inside them.

――― Circle 1

――― Circle 2 ――― Circle 3

Module 15 815 Lesson 4

15.4 Segment Relationships in Circles

Essential Question: What are the relationships between the segments in circles?

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C Fill in the chart with the lengths of the segments measured to the nearest millimeter and calculate their products.

DG GC EG GF DG ⋅ GC EG ⋅ GF

Circle 1

Circle 2

Circle 3

Look for a pattern among the measurements and calculations of the segments. From the table, it appears that will always equal .

Reflect

1. Discussion Compare your results with those of your classmates. What do you notice?

2. What conjecture can you make about the products of the segments of two chords that intersect inside a circle?

Conjecture:

Explain 1 Applying the Chord-Chord Product TheoremIn the Explore, you discovered a pattern in the relationship between the parts of two chords that intersect inside a circle. In this Example, you will apply the following theorem to solve problems.

Chord-Chord Product Theorem

If two chords intersect inside a circle, then the products of the lengths of the segments of the chords are equal.



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Example 1 Find the value of x and the length of each chord.

A Set up an equation according to the Chord-Chord Product Theorem and solve for x.

CE ⋅ ED = AE ⋅ EB

6 (2) = 3 (x)

12 = 3x

4 = x

Add the segment lengths to find the length of each chord.

CD = CE + ED = 6 + 2 = 8

AB = AE + EB = 4 + 3 = 7

B Set up an equation according to the Chord-Chord Product Theorem and solve for x:

HG ⋅ GJ = KG ⋅ GI

( ) = ( ) = 6x

= x

Add the segment lengths together to find the lengths of each chord:

HJ = HG + GJ = + 8 =

KI = + GI = 6 + =

Your Turn

3. Given AD = 12. Find the value of x and the length of each chord.


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Explain 2 Proving the Secant-Secant Product TheoremA secant is any line that intersects a circle at exactly two points. A secant segment is part of a secant line with at least one point on the circle. A secant segment that lies in the exterior of the circle with one point on the circle is called an external secant segment. Secant segments drawn from the same point in the exterior of a circle maintain a certain relationship that can be stated as a theorem.

Secant-Secant Product Theorem

If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

Example 2 Use similar triangles to prove the Secant-Secant Product Theorem.

Step 1 Identify the segments in the diagram. The whole secant segments in this

diagram are and .

The external secant segments in this diagram are and .

Step 2

Given the diagram as shown, prove that AE ⋅ BE = CE ⋅ DE.

Prove: AE ⋅ BE = CE ⋅ DE

Proof: Draw auxiliary line segments ̄ AD and ̄ CB . ∠EAD and ∠ECB both

intercept , so ∠ ≅ ∠ . ∠E ≅ ∠E by the

Property. Thus, △EAD ∼ △ECB by

the . Therefore, corresponding sides

are proportional, so AE _ = _ BE . By the Property of Equality,

BE (CE) ⋅ AE ___ CE = DE ___ BE ⋅ BE (CE) , and thus AE ⋅ BE = CE ⋅ DE.

Reflect

4. Rewrite the Secant-Secant Theorem in your own words. Use a diagram or shortcut notation to help you remember what it means.

5. Discussion: Suppose that two secants are drawn so that they intersect on the circle. Can you determine anything about the lengths of the segments formed? Explain.



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Explain 3 Applying the Secant-Secant Product TheoremYou can use the Secant-Secant Product Theorem to find unknown measures of secants and secant segments by setting up an equation.

Example 3 Find the value of x and the length of each secant segment.

A Set up an equation according to the Secant-Secant Product Theorem and solve for x.

AC ⋅ AB = AE ⋅ AD

(5 + x) (5) = (12) (6)

5x + 25 = 72

5x = 47

x = 9.4

Add the segments together to find the lengths of each secant segment.

AC = 5 + 9.4 = 14.4; AE = 6 + 6 = 12

B Set up an equation according to the Secant-Secant Product Theorem and solve for x.

UP ⋅ TP = SP ⋅ RP

( ) (7) = ( ) (6)

x + =

x =

x =

Add the segments together to find the lengths of each secant segment.

UP = 7 + = ; SP = 8 + 6 = 14

Your Turn

Find the value of x and the length of each secant segment.

6. 7.



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Explain 4 Applying the Secant-Tangent Product TheoremA similar theorem applies when both a secant segment and tangent segment are drawn to a circle from the same exterior point. A tangent segment is a segment of a tangent line with exactly one endpoint on the circle.

Secant-Tangent Product Theorem

If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared.

Example 4 Find the value of x.

A Given the diameter of the Earth as 8,000 miles, a satellite’s orbit is 6,400 miles above the Earth. Its range, shown by

_ SP , is a tangent segment.

Set up an equation according to the Secant-Tangent Product Theorem and solve for x:

SA ⋅ SE = SP 2

(8000 + 6400) (6400) = x 2

(14400) (6400) = x 2

92,160,000 = x 2

±9600 = x

Since distance must be positive, the value of x must be 9600 miles.

B Set up an equation according to the Secant-Tangent Product Theorem and solve for x:

BD ⋅ BC = BA 2

( ) (2) = 5 2

x + =

x =

x =



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Reflect

8. Compare and contrast the Secant-Secant Product Theorem with the Secant-Tangent Product Theorem.

Your Turn

Find the value of x.

9. On a bird-watching trip, you travel along a path tangent to a circular pond to a lookout station that faces a hawk’s nest. Given the measurements in the diagram on your bird-watching map, how far is the nest from the lookout station?

10.



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Elaborate

11. How is solving for y in the following diagram different from Example 3?

12. A circle is constructed with two secant segments that intersect outside the circle. If both external secant segments are equal, is it reasonable to conclude that both secant segments are equal? Explain.

13. Essential Question Check-In How are the theorems in this lesson related?



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• Online Homework• Hints and Help• Extra Practice

Use the figure for Exercises 1−2.

Suppose you use geometry software to construct two chords ̄ RS and ̄ TU that intersect inside a circle at V.

1. If you measured _ RV , _ VS , _ TV , and

_ VU , what

would be true about the relationship between their lengths?

2. Suppose you drag the points around the circle and examine the changes in the measurements. Would your answer to Exercise 1 change? Explain.

Use the figure for Exercises 3–4.

Suppose you use geometry software to construct two secants

‹ − › DC and ‹ − › BE that

intersect outside a circle at F.

3. If you measured _ DF , _ CF , _ BF , and

_ EF , what

would be true about the relationship between their lengths?

4. Suppose you drag F and examine the changes in the measurements. Would your answer to Exercise 3 change? Explain.

Find the value of the variable and the length of each chord.

5. 6.

Evaluate: Homework and Practice



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7. M is the midpoint of _ PQ , The diameter of circle O is 13 in. and RM = 4 in.

a. Find PM.

b. Find PQ.

8. Representing a Real-World Problem A broken pottery shard found at archaeological dig has a curved edge. Find the diameter of the original plate. (Use the fact that the diameter

_ PR is the

perpendicular bisector of chord _ AB .)

9. Critique Reasoning A student drew a circle and two secant segments. He concluded that if

_ PQ ≅

_ PS , then

_ QR ≅

_ ST . Do you agree with the

student’s conclusion? Why or why not?

Find the value of the variable and the length of each secant segment.

10. 11.



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12. Find the value of x.

Find the value of the variable.

13. 14.

15.

16. Tangent ‹ − › PF and secants ‹ − › PD and ‹ − › PB are drawn to circle A. Determine whether each of the following relationships is true or false. Select the correct answer for each lettered part.a. PB · EB = PD · DC True Falseb. PE · EB = PC · DC True Falsec. PB · PE = PF 2 True Falsed. PB · DC = PD · EB True Falsee. PB · PE = PD · PC True Falsef. PB · PE = PF · PC True False

17. Which of these is closest to the length of tangent segment _ PQ ?

A. 6.9 B. 9.2C. 9.9 D. 10.6



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18. Explain the Error Below is a student’s work to find the value of x. Explain the error and find the correct value of x.

AB ⋅ BC = D C 2

6 (4) = x 2

x 2 = 24

x = ± √_

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6

19. Represent Real-World Problems Molokini is a small, crescent-shaped island 2 1 _ 2 miles from the Maui, Hawaii, coast. It is all that remains of an extinct volcano. To approximate the diameter of the mouth of the volcano, a geologist used a diagram like the one shown. The geologist assumed that the mouth of the volcano was a circle. What was the approximate diameter of the volcano’s mouth to the nearest ten feet?

20. Multi-step Find the value of both variables in the figure.

21. _ KL is a tangent segment of circle N and

_ KM and

_ LM are secants of the circle.

Find the value of x.

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H.O.T. Focus on Higher Order Thinking

22. Justify Reasoning Prove the Chord-Chord Product Theorem

Given: Chords _ AB and

_ CD intersect at point E.

Prove: AE ⋅ EB = CE ⋅ ED (Hint: Draw _ AC and

_ BD .)

23. Justify Reasoning _ PQ is a tangent segment of a circle with radius 4 in.

Q lies on the circle, and PQ = 6 in. Make a sketch and find the distance from P to the circle. Round to the nearest tenth of an inch. Explain your reasoning.

24. Justify Reasoning The circle in the diagram has radius c. Use this diagram and the Chord-Chord Product Theorem to prove the Pythagorean Theorem.

25. Critical Thinking The radius of circle A is 4. CD = 4, and _ CB is a tangent

segment. Describe two different methods you can use to find BC.



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Lesson Performance Task

The figure shows the basic design of a Wankel rotary engine. The triangle is equilateral, with sides measuring 10 inches. An arc on each side of the triangle has as its center the vertex on the opposite side of the triangle. In the figure, the arc ADB is an arc of a circle with its center at C.

a. Use the sketch of the engine. What is the measure of each arc along the side of the triangle?

b. Use the relationships in an equilateral triangle to find the value of x. Explain.

c. Use the Chord-Chord Product Theorem to find the value of x. Explain.



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