Similarity in Right Triangles

1
Similarity in Right Triangles

• 10.3
• Finding relationships among lengths of sides of a
  right triangle and the altitude to the hypotenuse

2
Geometric Mean

• Proportions in which the means are equal occur
  frequently in geometry. For any two positive
  numbers a and b, the geometric mean of a and b is
  the positive number such that a/xx/b

3
Example I

• Find the geometric mean of 4 and 18
• 4/xx/18
• x272
• xv72
• x6v2

4
Theorem 10.3

• 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

c
a
b
d
5
Corollary 1

• 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/CDCD/DB

c
b
a
d
6
Corollary 2

• The altitude to the hypotenuse of a right
  triangle intersects it so that the length of each
  leg is the geometric mean of the length of its
  adjacent segment of the hypotenuse and the length
  of the entire hypotenuse
• AB/ACAC/AC
• AB/CBCB/DB

c
b
a
d
7
Example II Solve for x and y
x
y
4
5

• Use Corollary 1 to solve for x
• 45/x x/4
• x2 36
• x 6

• Use Corollary 2 to solve for y
• 4/yy/5
• y2 20
• y 4v5

8
Example III

• At the parking lot of a state park, the 300m path
  to the snack bar and the 400m path to the boat
  rental shop meet at a right angle. Marla walks
  straight from the parking lot to the ocean. How
  far is Marla from the snack bar?

Snack Bar
Parking
Marla
Boat rental
9
Example III Cont
Snack Bar

• AD/AC AC/AB
• (Corollary II)
• AD/300 300/500
• Substitute
• 500(AD) 90,000
• (Cross Product)
• AD 180
• (Divide by 500)

A
300m
Parking
Marla
C
D
400m
Boat rental
B
10
Homework

• P 514 1-19, 24-27