<meta http-equiv="refresh" content="1; url=/nojavascript/"> Similar Triangles Review | CK-12 Foundation
Dismiss
Skip Navigation
You are reading an older version of this FlexBook® textbook: CK-12 Foundation and Leadership Public Schools, College Access Reader: Geometry Go to the latest version.

[^*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

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

Description

Authors:

Grades:

8 , 9 , 10

Date Created:

Feb 23, 2012

Last Modified:

Dec 12, 2013
You can only attach files to None which belong to you
If you would like to associate files with this None, please make a copy first.

Reviews

Please wait...
You need to be signed in to perform this action. Please sign-in and try again.
Please wait...
Image Detail
Sizes: Medium | Original
 

Original text