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Segments from Secants

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What if you were given a circle with two secants that intersect outside the circle? How could you use the length of some of the segments formed by their intersection to determine the lengths of the unknown segments? After completing this Concept, you'll be able to use the Two Secants Segments Theorem to solve problems like this one.

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Segments from Secants CK-12

Guidance

When two secants intersect outside a circle, the circle divides the secants into segments that are proportional with each other.

Two Secants Segments Theorem: If two secants are drawn from a common point outside a circle and the segments are labeled as below, then a(a+b)=c(c+d) .

Example A

Find the value of x .

Use the Two Secants Segments Theorem to set up an equation.

18 \cdot (18+x) &= 16 \cdot (16+24)\\324+18x &= 256+384\\18x &= 316\\x &= 17\frac{5}{9}

Example B

Find the value of x .

Use the Two Secants Segments Theorem to set up an equation.

x \cdot (x+x) &= 9 \cdot 32\\2x^2 &= 288\\x^2 &= 144\\x &=12, \ x\ne -12 \ (\text{length is not negative})

Example C

True or False: Two secants will always intersect outside of a circle.

This is false. If the two secants are parallel, they will never intersect. It's also possible for two secants to intersect inside a circle.

Segments from Secants CK-12

Guided Practice

Find x in each diagram below. Simplify any radicals.

1.

2.

3.

Answers:

Use the Two Secants Segments Theorem.

1.

8(8+x)&=6(6+18)\\64+8x&=144\\8x&=80\\x&=10

2.

4(4+x)&=3(3+13)\\16+4x&=48\\4x&=32\\x&=8

3.

15(15+27)&=x\cdot 45\\630&=45x\\x&=14

Practice

Fill in the blanks for each problem below. Then, solve for the missing segment.

3(\underline{\;\;\;\;\;\;\;}+\underline{\;\;\;\;\;\;\;})=2(2+7)

x \cdot \underline{\;\;\;\;\;\;\;}=8(\underline{\;\;\;\;\;\;\;}+\underline{\;\;\;\;\;\;\;})

Find x in each diagram below. Simplify any radicals.

  1. Fill in the blanks of the proof of the Two Secants Segments Theorem.

Given : Secants \overline{PR} and \overline{RT}

Prove : a(a+b)=c(c+d)

Statement Reason
1. Secants \overline{PR} and \overline{RT} with segments a, \ b, \ c, and d . 1. Given
2. \angle R \cong \angle R 2. Reflexive PoC
3. \angle QPS \cong \angle STQ 3. Congruent Inscribed Angles Theorem
4. \triangle RPS \sim \triangle RTQ 4. AA Similarity Postulate
5. \frac{a}{c+d}=\frac{c}{a+b} 5. Corresponding parts of similar triangles are proportional
6. a(a+b)=c(c+d) 6. Cross multiplication

Solve for the unknown variable.

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