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# Segments from Secants

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What if you wanted to figure out the distance from the orbiting moon to different locations on Earth? At a particular time, the moon is 238,857 miles from Beijing, China. On the same line, Yukon is 12,451 miles from Beijing. Drawing another line from the moon to Cape Horn (the southernmost point of South America), we see that Jakarta, Indonesia is collinear. If the distance from Cape Horn to Jakarta is 9849 miles, what is the distance from the moon to Jakarta? After completing this Concept, you'll be able to solve problems like this.

### Guidance

In addition to forming an angle outside of a circle, the circle can divide the secants into segments that are proportional with each other.

If we draw in the intersecting chords, we will have two similar triangles.

From the inscribed angles and the Reflexive Property $( \angle R \cong \angle R), \triangle PRS \sim \triangle TRQ$ . Because the two triangles are similar, we can set up a proportion between the corresponding sides. Then, cross-multiply. $\frac{a}{c+d}=\frac{c}{a+b} \Rightarrow a(a+b)=c(c+d)$

Two Secants Segments Theorem: If two secants are drawn from a common point outside a circle and the segments are labeled as above, then $a(a+b)=c(c+d)$ . In other words, the product of the outer segment and the whole of one secant is equal to the product of the outer segment and the whole of the other secant.

#### Example A

Find the value of the missing variable.

Use the Two Secants Segments Theorem to set up an equation. For both secants, you multiply the outer portion of the secant by the whole.

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

#### Example B

Find the value of the missing variable.

Use the Two Secants Segments Theorem to set up an equation. For both secants, you multiply the outer portion of the secant by the whole.

$x \cdot (x+x)=9 \cdot 32\\2x^2=288\\x^2=144\\x=12$

$x \neq -12$ because length cannot be 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.

Watch this video for help with the Examples above.

#### Concept Problem Revisited

The given information is to the left. Let’s set up an equation using the Two Secants Segments Theorem.

$238857 \cdot 251308 &= x \cdot (x+9849)\\60026674956 &= x^2+9849x\\0 &= x^2+9849x-60026674956\\Use \ the \ Quadratic \ Formula \ x & \approx \frac{-9849 \pm \sqrt{9849^2-4(-60026674956)}}{2}\\x & \approx 240128.4 \ miles$

### Vocabulary

A circle is the set of all points that are the same distance away from a specific point, called the center . A radius is the distance from the center to the circle. A chord is a line segment whose endpoints are on a circle. A diameter is a chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. A central angle is the angle formed by two radii and whose vertex is at the center of the circle. An inscribed angle is an angle with its vertex on the circle and whose sides are chords. The intercepted arc is the arc that is inside the inscribed angle and whose endpoints are on the angle. A tangent is a line that intersects a circle in exactly one point. The point of tangency is the point where the tangent line touches the circle. A secant is a line that intersects a circle in two points.

### 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

Solve for the missing segment.

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

1. Prove the Two Secants Segments Theorem.

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

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

Solve for the unknown variable.