Lesson 7.3 ­ Similarity in Right Triangles.notebook
November 07, 2016
Lesson 7.3
Similarity in Right Triangles
Objective: To use properties of the altitude of a right triangle. The geometric mean between 2 positive numbers (a and b) is x such that:
a x =
x b
Geometric mean = x
*x is the denominator of the first ratio and the numerator of the second ratio*
Examples:
1) Find the geometric mean between 2 and 10.
2) Find the geometric mean between 5 and 25.
3) Find the geometric mean between √3 and √3.
1
Lesson 7.3 ­ Similarity in Right Triangles.notebook
November 07, 2016
Theorem 7.5:
The altitude of a right triangle forms two triangles that are similar to the given triangle and to each other.
C
B
A
D
What triangles are similar? ex: Find x, y, z A
9
B
** remember these 3 triangles are similar because of AA!
16
y
x
z
D
C
ΔDBC
ΔABD
ΔADC
Short leg
Long Leg
Hypotenuse
ex: Find x, y, z A
5
B
8
y
x
D
ΔABD
z
ΔDBC
C
ΔADC
Short leg
Long Leg
Hypotenuse
2
Lesson 7.3 ­ Similarity in Right Triangles.notebook
November 07, 2016
Geometric Mean (altitude) Theorem:
The altitude of a right triangle is the geometric mean between
the parts (segments) of the hypotenuse.
Pt hyp 1 Alt. =
Alt Pt. hyp 2
A
D
B
C
Alt = CD
Whole hyp = AB
Pt. hyp1 = AD
Pt. hyp 2 = DB
∴ AD CD or DB CD =
=
CD DB CD AD
Examples:
1) Find x
5
8
x
2) Find x
x
20
10
Geometric Mean (leg) Theorem:
The leg of a right triangle is the geometric mean between
the whole hypotenuse and the part adjacent to the leg.
A
leg = AC
leg = BC
alt = CD
Wh. hyp = AB Pt. hyp. Adj to AC = AD
pt. hyp Adj to BC = DB
D
C
B
and AB BC ∴ AB AC =
=
AC AD BC DB
3
Lesson 7.3 ­ Similarity in Right Triangles.notebook
Example:
Find AB and BC
November 07, 2016
C
5
8
A
B
4