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?

### Segments from Secants

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 \begin{align*}( \angle R \cong \angle R), \triangle PRS \sim \triangle TRQ\end{align*}. Because the two triangles are similar, we can set up a proportion between the corresponding sides. Then, cross-multiply. \begin{align*}\frac{a}{c+d}=\frac{c}{a+b} \Rightarrow a(a+b)=c(c+d)\end{align*}

**Two Secants Segments Theorem:** If two secants are drawn from a common point outside a circle and the segments are labeled as above, then \begin{align*}a(a+b)=c(c+d)\end{align*}. 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.

#### Applying the Two Secants Segments Theorem

1. 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.

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

2. 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.

\begin{align*}x \cdot (x+x)=9 \cdot 32\\ 2x^2=288\\ x^2=144\\ x=12\end{align*}

\begin{align*}x \neq -12\end{align*} because length cannot be negative.

#### Understanding Properties of Secants

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.

#### Earlier Problem Revisited

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

\begin{align*}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\end{align*}

### Examples

Find \begin{align*}x\end{align*} in each diagram below. Simplify any radicals.

#### Example 1

Use the Two Secants Segments Theorem.

\begin{align*}8(8+x)&=6(6+18)\\64+8x&=144\\8x&=80\\x&=10\end{align*}

#### Example 2

\begin{align*}4(4+x)&=3(3+13)\\16+4x&=48\\4x&=32\\x&=8\end{align*}

#### Example 3

\begin{align*}15(15+27)&=x\cdot 45\\630&=45x\\x&=14\end{align*}

### Interactive Practice

### Review

Solve for the missing segment.

Find \begin{align*}x\end{align*} in each diagram below. Simplify any radicals.

- Prove the Two Secants Segments Theorem.

Given: Secants \begin{align*}\overline{PR}\end{align*} and \begin{align*}\overline{RT}\end{align*}

Prove: \begin{align*}a(a+b)=c(c+d)\end{align*}

Solve for the unknown variable.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 9.10.