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# Segments from Secants

## Relationships of products of sections versus sums of sections of secants.

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Segments from Secants

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.

### 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.

### Guided Practice

Find $x$ in each diagram below. Simplify any radicals.

1.

2.

3.

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.

### Vocabulary Language: English Spanish

central angle

central angle

An angle formed by two radii and whose vertex is at the center of the circle.
chord

chord

A line segment whose endpoints are on a circle.
circle

circle

The set of all points that are the same distance away from a specific point, called the center.
diameter

diameter

A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
Inscribed Angle

Inscribed Angle

An inscribed angle is an angle with its vertex on the circle. The measure of an inscribed angle is half the measure of its intercepted arc.
intercepted arc

intercepted arc

The arc that is inside an inscribed angle and whose endpoints are on the angle.
point of tangency

point of tangency

The point where the tangent line touches the circle.

The distance from the center to the outer rim of a circle.
AA Similarity Postulate

AA Similarity Postulate

If two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar.
Congruent

Congruent

Congruent figures are identical in size, shape and measure.
Reflexive Property of Congruence

Reflexive Property of Congruence

$\overline{AB} \cong \overline{AB}$ or $\angle B \cong \angle B$
Secant

Secant

The secant of an angle in a right triangle is the value found by dividing length of the hypotenuse by the length of the side adjacent the given angle. The secant ratio is the reciprocal of the cosine ratio.
secant line

secant line

A secant line is a line that joins two points on a curve.
Tangent line

Tangent line

A tangent line is a line that "just touches" a curve at a single point and no others.
Two Secants Segments Theorem

Two Secants Segments Theorem

Two secants segments theorem states that if you have a point outside a circle and draw two secant lines from it, there is a relationship between the line segments formed.