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# 9.5: Similar Triangles Review

Difficulty Level: At Grade Created by: CK-12

[$^*$Editor’s note: This day is set aside for a quiz and a review of similar triangles.]

## Learning Objectives

• Review the concepts of angle congruence and segment proportionality in similar triangles.

## Chords

Remember from lesson 2: a chord is a line segment that has both endpoints on a circle.

• Line segments whose endpoints are both on a circle are called _______________.

Segments of Chords Theorem

If two chords intersect inside a circle so that one chord is divided into segments of lengths $a$ and $b$ and the other into segments of lengths $c$ and $d$, then the segments of the chords satisfy the following relationship:

$ab = cd$

This means that the product of the segment lengths of one chord equals the product of the segment lengths of the second chord:

• The intersection point splits each ______________________ into two segments.
• The product of both segment lengths of one chord is ____________________ to the product of both segment lengths of the other chord.

We prove this theorem on the following page.

Proof

We connect points $A$ and $C$ and points $D$ and $B$ to make $\Delta AEC$ and $\Delta DEB$:

Statements Reasons
1. $\angle AEC \cong \angle DEB$ 1. Vertical angles are congruent
2. $\angle CAB \cong \angle BDC$ 2. Inscribed angles intercept the same arc
3. $\angle ACD \cong \angle ABD$ 3. Inscribed angles intercept the same arc
4. $\Delta AEC \cong \Delta DEB$ 4. AA similarity postulate
5. $\frac{c}{b} = \frac{a}{d}$ 5. In similar triangles, the ratios of corresponding sides are equal.
6. $ab = cd$ 6.Cross multiplication

Example 1

Find the value of the variable $x$:

Use the products of the segment lengths of each chord:

$10x & = 8 \cdot 12\\10x & = 96\\x & = 9.6$

1. In your own words, define a chord.

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2. True or false: When two chords intersect inside a circle, the sum of the segment lengths of one chord is equal to the sum of the segment lengths of the other chord.

3. How could you change the statement in #2 above to make it true?

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4. In the space below, make up your own problem with two chords that intersect inside a circle, and then solve your problem.

(Hint: if you are having trouble, look at Example 1 on the previous page and model your problem after that one.)

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## Date Created:

Feb 23, 2012

May 12, 2014
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