9.5: Similar Triangles Review
[\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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