7-4 Similarity in Right Triangles
Theorem 7-3: Similarity in Right Triangles Theorem
 The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are
similar to the original triangle and to each other.
 Write a similarity statement about the three triangles in this diagram:
Geometric Mean
 Geometric mean is a concept that is used to indicate one type of central tendency.
 The geometric mean of two positive numbers, a and b, is the length of one side of a square
whose area is equal to the area of a rectangle with sides of lengths a and b.

For any two positive numbers a and b, the geometric mean is the positive number x such that
𝑎
𝑥
=
𝑥
𝑏

Note: What does x = ?
Geometric Mean: Practice
 Calculate the geometric mean of 15 and 20.

Calculate the geometric mean of 8 and 12.

Calculate the geometric mean of 9 and 24.

Calculate the geometric mean of 7 and 9.
Corollary 1 to Theorem 7-3:
 The length of the altitude to the
hypotenuse of a right triangle is the
geometric mean of the lengths of the
segments of the hypotenuse.

𝐴𝐷

𝐵𝐷 = √𝐴𝐷 ∙ 𝐷𝐶

How can this corollary be proven?
𝐵𝐷
𝐵𝐷
= 𝐷𝐶
Using Corollary 1
 Solve for y
Corollary 2 to Theorem 7-3
 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 adjacent
hypotenuse segment and the length of
the hypotenuse.

𝐴𝐷

𝐴𝐵 = √𝐴𝐷 ∙ 𝐴𝐶, 𝐵𝐶 = √𝐴𝐶 ∙ 𝐷𝐶

How can this corollary be proven?
𝐴𝐵
=
𝐴𝐵
𝐴𝐶
,
𝐷𝐶
𝐵𝐶
𝐵𝐶
= 𝐴𝐶
Using Corollary 2
 Solve for x
Using the corollaries
 Solve for x, y and z
Using the corollaries
 Ms. Vargas’ car broke down
at the top of the Hwy 92 exit
(the blue dot). She’s going to
a building at the corner of 2nd
St. and 3rd Ave.
 About how far will she have
to walk to get to her
destination?

About how far would it be if she could fly?