Prentice Hall Geometry

Chapter 7 Similarity > 7-4 Similarity in Right Triangles

Take note Corollary 2 to Theorem 7-3

Corollary

The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to the leg.

If …

Triangle ABC has altitude line from right angle C to D on side AB.

Then …

A B A C = A C A D A B C B = C B D B table with 2 rows and 1 column , row1 column 1 , fraction eh b , over eh c end fraction . equals . fraction eh c , over eh d end fraction , row2 column 1 , fraction eh b , over c b end fraction . equals . fraction c b , over d b end fraction , end table

Example

A right triangle is divided by altitude into other right triangles.
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You will prove Corollary 2 in Exercise 43.

The corollaries to Theorem 7-3 give you ways to write proportions using lengths in right triangles without thinking through the similar triangles. To help remember these corollaries, consider the diagram and these properties.

A right triangle is divided into two smaller right triangles by an altitude line.
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Corollary 1

s 1 a = a s 2 fraction s sub 1 , over eh end fraction , equals , fraction eh , over s sub 2 end fraction

Corollary 2

h l 1 = l 1 s 1 , h l 2 = l 2 s 2 fraction h , over l sub 1 end fraction , equals . fraction l sub 1 , over s sub 1 end fraction . comma , fraction h , over l sub 2 end fraction , equals . fraction l sub 2 , over s sub 2 end fraction

Problem 3 Using the Corollaries

Algebra What are the values of x and y?

A right triangle with a leg measuring x has altitude line measuring y dividing the hypotenuse into segments measuring 12 and 4, forming a triangle with legs y and 4 and hypotenuse x.

Use Corollary 2. ► 4 + 12 x = x 4 Write a proportion . 4 y = y 12 ◄ Use Corollary 1. x 2 = 64 Cross Products Property y 2 = 48 x = 64 Take the positive square root . y = 48 x = 8 Simplify . y = 4 3 table with 4 rows and 5 columns , row1 column 1 , begin box , cap usecap corollary . 2. , end box . u 2 . fraction 4 plus 12 , over x end fraction , column 2 equals , x over 4 , column 3 cap writeaproportion . . , column 4 4 over y , column 5 equals , y over 12 , black left pointing pointer . begin box , cap usecap corollary . 1. , end box , row2 column 1 , x squared , column 2 equals 64 , column 3 cap crosscap productscap property , column 4 y squared , column 5 equals 48 , row3 column 1 , x , column 2 equals square root of 64 , column 3 cap takethepositivesquareroot . . , column 4 y , column 5 equals square root of 48 , row4 column 1 , x , column 2 equals 8 , column 3 cap simplify , . , column 4 y , column 5 equals 4 square root of 3 , end table

✓ Got It?

What are the values of x and y?

End ofPage 463

Take note Corollary 2 to Theorem 7-3

Problem 3 Using the Corollaries

Plan

✓ Got It?

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