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# Segments from Chords

## Products of the segments of each of two intersecting chords are equal.

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Segments from Chords

What if Ishmael wanted to know the diameter of a CD from his car? He found a broken piece of one in his car and took some measurements. He places a ruler across two points on the rim, and the length of the chord is 9.5 cm. The distance from the midpoint of this chord to the nearest point on the rim is 1.75 cm. Find the diameter of the CD. After completing this Concept, you'll be able to use your knowledge of chords to solve this problem.

### Guidance

When two chords intersect inside a circle, the two triangles they create are similar, making the sides of each triangle in proportion with each other. If we remove AD¯¯¯¯¯¯¯¯\begin{align*}\overline{AD}\end{align*} and BC¯¯¯¯¯¯¯¯\begin{align*}\overline{BC}\end{align*} the ratios between AE¯¯¯¯¯¯¯¯,EC¯¯¯¯¯¯¯¯,DE¯¯¯¯¯¯¯¯\begin{align*}\overline{AE}, \overline{EC}, \overline{DE}\end{align*}, and EB¯¯¯¯¯¯¯¯\begin{align*}\overline{EB}\end{align*} will still be the same.

Intersecting Chords Theorem: If two chords intersect inside a circle so that one is divided into segments of length a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*} and the other into segments of length c\begin{align*}c\end{align*} and d\begin{align*}d\end{align*} then ab=cd\begin{align*}ab = cd\end{align*}. In other words, the product of the segments of one chord is equal to the product of segments of the second chord.

#### Example A

Find x\begin{align*}x\end{align*} in the diagram below.

Use the ratio from the Intersecting Chords Theorem. The product of the segments of one chord is equal to the product of the segments of the other.

128=10x96=10x9.6=x\begin{align*}12 \cdot 8=10 \cdot x\\ 96=10x\\ 9.6=x\end{align*}

#### Example B

Find x\begin{align*}x\end{align*} in the diagram below.

Use the ratio from the Intersecting Chords Theorem. The product of the segments of one chord is equal to the product of the segments of the other.

x15=5915x=45x=3\begin{align*}x \cdot 15=5 \cdot 9\\ 15x=45\\ x=3\end{align*}

#### Example C

Solve for x\begin{align*}x\end{align*}.

a)

b)

Again, we can use the Intersecting Chords Theorem. Set up an equation and solve for x\begin{align*}x\end{align*}.

a) 824=(3x+1)12192=36x+12180=36x5=x\begin{align*}8 \cdot 24=(3x+1) \cdot 12\\ 192=36x+12\\ 180=36x\\ 5=x\end{align*}

b) 3221=(x9)(x13)672=x222x+1170=x222x5550=(x37)(x+15)x=37,15\begin{align*}32 \cdot 21=(x-9)(x-13)\\ 672=x^2-22x+117\\ 0=x^2-22x-555\\ 0=(x-37)(x+15)\\ x=37, -15\end{align*}

However, x15\begin{align*}x \neq -15\end{align*} because length cannot be negative, so x=37\begin{align*}x=37\end{align*}.

Watch this video for help with the Examples above.

#### Concept Problem Revisited

Think of this as two chords intersecting each other. If we were to extend the 1.75 cm segment, it would be a diameter. So, if we find x\begin{align*}x\end{align*} in the diagram below and add it to 1.75 cm, we would find the diameter.

4.254.2518.0625x=1.75x=1.75x10.3 cm, making the diameter10.3+1.75 12 cm, which is theactual diameter of a CD.\begin{align*}4.25 \cdot 4.25&=1.75\cdot x\\ 18.0625&=1.75x\\ x & \approx 10.3 \ cm,\ \text{making the diameter} 10.3 + 1.75 \approx \ 12 \ cm, \ \text{which is the}\\ & \qquad \qquad \qquad \text{actual diameter of a CD.}\end{align*}

### 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.

### Guided Practice

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

1.

2.

3.

For all problems, use the Intersecting Chords Theorem.

1.

15460x=5x=5x=12\begin{align*}15\cdot 4 &=5\cdot x\\ 60&=5x \\ x&=12\end{align*}

2.

18x18xx=93=27=1.5\begin{align*}18 \cdot x &=9\cdot 3\\18x &=27\\ x&=1.5\end{align*}

3.

12x12xx=916=144=12\begin{align*} 12 \cdot x &=9 \cdot 16 \\ 12x&=144\\ x&=12\end{align*}

### Practice

1. If two chords bisect one another then they are diameters.
2. Tangent lines can create chords inside circles.
3. If two chords intersect and you know the length of one chord, you will be able to find the length of the second chord.

Solve for the missing segment.

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

Find the value of x\begin{align*}x\end{align*}.

1. Suzie found a piece of a broken plate. She places a ruler across two points on the rim, and the length of the chord is 6 inches. The distance from the midpoint of this chord to the nearest point on the rim is 1 inch. Find the diameter of the plate.
2. Prove the Intersecting Chords Theorem.

Given: Intersecting chords AC¯¯¯¯¯¯¯¯\begin{align*}\overline{AC}\end{align*} and BE¯¯¯¯¯¯¯¯\begin{align*}\overline{BE}\end{align*}.

Prove: ab=cd\begin{align*}ab=cd\end{align*}

### 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.
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 angle with its vertex on the circle and whose sides are chords.
intercepted arc

intercepted arc

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

Intersecting Chords Theorem

According to the Intersecting Chords Theorem, if two chords intersect inside a circle so that one is divided into segments of length a and b and the other into segments of length c and d, then ab = cd.