9.5: Similar Triangles Review
[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 and and the other into segments of lengths and , then the segments of the chords satisfy the following relationship:
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 and and points and to make and :
Statements | Reasons |
---|---|
1. | 1. Vertical angles are congruent |
2. | 2. Inscribed angles intercept the same arc |
3. | 3. Inscribed angles intercept the same arc |
4. | 4. AA similarity postulate |
5. | 5. In similar triangles, the ratios of corresponding sides are equal. |
6. | 6.Cross multiplication |
Example 1
Find the value of the variable :
Use the products of the segment lengths of each chord:
Reading Check:
1. In your own words, define a chord.
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?
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.)
Image Attributions
Concept Nodes:
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