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9.5: Similar Triangles Review

Difficulty Level: At Grade Created by: CK-12
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[\begin{align*}^*\end{align*}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 \begin{align*}a\end{align*} and \begin{align*}b\end{align*} and the other into segments of lengths \begin{align*}c\end{align*} and \begin{align*}d\end{align*}, then the segments of the chords satisfy the following relationship:

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

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 \begin{align*}A\end{align*} and \begin{align*}C\end{align*} and points \begin{align*}D\end{align*} and \begin{align*}B\end{align*} to make \begin{align*}\Delta AEC\end{align*} and \begin{align*}\Delta DEB\end{align*}:

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

Example 1

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

Use the products of the segment lengths of each chord:

\begin{align*}10x & = 8 \cdot 12\\ 10x & = 96\\ x & = 9.6\end{align*}

Reading Check:

1. In your own words, define a chord.

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

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

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?

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

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

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.)

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

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

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8 , 9 , 10
Date Created:
Feb 23, 2012
Last Modified:
May 12, 2014
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