CC Geometry H
Aim #18: How does the altitude drawn to the hypotenuse of a right triangle
create similar triangles?
Do Now: The following right triangles are similar because of _____________.
Find the values of x and y.
3
2
2
x
y
An altitude of a triangle is a segment from a vertex ____________________ to
the opposite side. BD is the altitude from vertex _____ to side _____.
A
A
D
D
B
C
B
1. Explain how ΔABC ~ ΔBDC. Be specific.
2. Explain how ΔABC ~ ΔADB. Be specific.
3. Since ΔABC ~ ΔBDC and ΔABC ~ ΔADB, then ___________ ~ ___________.
The conclusion in (3) supports the fact that similarity of triangles has the
_____________________________ property.
We have shown the three triangles are similar: ΔABC ~ ΔBDC ~ ΔADB
General conclusion:
An altitude drawn from the right angle of a right triangle to the hypotenuse
divides the triangle into two similar sub-triangles, which are also similar to
the original triangle.
C
Move the three triangles to show they are similar.
large Δ
medium Δ small Δ
ΔABC ~ ΔBDC ~ΔADB
A
D
B
C
Given right ΔABC with altitude BD drawn to side AC. a) Mark the congruent
corresponding angles in the 3 triangles from our previous similarity statement:
B
ΔABC ~ ΔBDC ~ΔADB
b) Label the sides of each triangle
as L1, L2, and H.
A
C
D
Given ΔABC with sides 5, 12, and 13, altitude BD drawn to side AC.
B
large Δ
medium Δ
small Δ
L1
5
A
12
z
x
L2
y
D
13
C
H
Remembering that corresponding sides of similar triangles have the same ratio,
write and solve a proportion to determine the length of x.
Write and solve a proportion to determine the length of y.
Write and solve a proportion to determine the length of z.
Practice
1. Given ΔEFG with altitude FH drawn to the hypotenuse. a) Mark the congruent
corresponding angles in the 3 triangles. b) Complete the similarity statement:
G
ΔEFG ~ __________~___________
c) Complete the chart:
large Δ
20
H
16
medium Δ
small Δ
L1
L2
H
E
12
F
d) find lengths EH, FH, and GH:
2. Given right ΔIMJ with altitude JL, JL = 32, and IL = 24, find IJ, JM, LM, and
IM.
J
large Δ
L1
32
I
24
L
L2
M
H
medium Δ
small Δ
3. Use similar triangles to find the length of the altitudes x and y.
A
a)
b) E
x
C
4
D
9
B
16
y
G
4
H
F
5. Find QR:
4. Find x:
Let's Sum it Up!
When an altitude is drawn to the hypotenuse of a given right triangle, it forms
two similar sub-triangles that are similar to the given right triangle.
The ratios of corresponding sides of similar triangles are equivalent and can be
used to find unknown lengths of a triangle.
Name __________________
Date __________________
CC Geometry H
HW #18
1. ΔRST has altitude SU drawn to hypotenuse RT, ST = 15, RS = 36, RT = 39.
a) Complete the similarity statement: ΔRST ~ Δ______ ~ Δ______
large Δ
b) Find lengths SU, TU, and RU.
medium Δ
small Δ
L1
L2
H
2. Use similar triangles to find the length of altitude z.
I
25
K 4 J
z
L
OVER
3. Given right triangle ABC with altitude CD, find AD, BD, AB, and DC.
C
2√5
√5
A
D
B
4. In right triangle ABD, AB = 53, and altitude DC = 14. Find the lengths of BC
and AC.
5. Find a, b, and x:
Mixed Review:
Simplify the following radicals:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)