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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 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*}

### Review

Solve for the missing segment.

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

1. 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.

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### Vocabulary Language: English Spanish

central angle

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

chord

A line segment whose endpoints are on a circle.

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

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

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

point of tangency

The point where the tangent line touches the circle.

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 figures are identical in size, shape and measure.

Reflexive Property of Congruence

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

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

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

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 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.