Download Similar Triangles

Aim: What is special about similar
triangles?
Do Now:
U
10 cm
R
In the diagram
at right
 PQR ~  STU.
X
15 cm
9 cm
P
12 cm
T
Y
S
Q
Name the pairs of corresponding angles:
Q&T
R&U
P&S
Aim: Similar Triangles
Course: Applied Geo.
Short cuts, anyone?
Problem #1
In the diagram
at right
 PQR ~  STU.
U
10 cm
R
X
15 cm
9 cm
P
12 cm
T
Y
S
Q
A. Name the pairs of corresponding angles:
Q&T
R&U
P&S
B. Name the pairs of corresponding sides:
PQ & ST QR & TU PR & SU
C. Find the ratio of similitude QR
= 15 = 3
between PQR and STU.
TU
10 2
D. Find the value of y. E. Find the value of x.
3 = 12 3y = 24
3 = 9 3x = 18
y
=
8
2
y
Aim: Similar Triangles
= 6Geo.
2
x Course:xApplied
Short cuts, anyone?
Similar Triangles
Theorem 1:
If the corresponding sides of two
triangles are in proportion, the triangles are
similar.
Note: this is only true for triangles!!
Theorem 2:
If two angles of one triangle are
congruent to two angles of a second triangle,
then the two triangles are similar.
Two triangles are similar if
AA  AA
Aim: Similar Triangles
Course: Applied Geo.
Model Problem
a. Explain why the two triangles are similar.
E
A
410
AA  AA
790
B
W
410
600
790
M
600
C
b. Name the three pair of corresponding sides.
AB & EM, BC & MW, AC & EW
c. Name the three pair of corresponding angles.
A &  E,  B &  M,  C &  W
Aim: Similar Triangles
Course: Applied Geo.
Model Problem
Determine if the two triangles are similar.
P
G
0.8 in
1.9 in
Q
0.4 in
1.3 in
R
1.0 in
H 0.7 in I
Since no angles are given we must
determine if the sides are in proportion.
PQ 0.8

2
GH 0.4
QR 1.3

 1.857
HI 0.7
Because we have shown that two sides of the
triangles are not in proportion, it is enough
then, toAim:state
that
Course: Applied Geo.
Similar Triangles
they are not similar.
Model Problem
Explain why the triangles are similar
W
R
450
450
V
S
B
WSR  VSB because vertical
angles are congruent
R  V because their measures
are equal
RSW  VSB because triangles
are similar is two angles of the
triangles are congruent AA  AA
Aim: Similar Triangles
Course: Applied Geo.
Model Problem
The lengths, in meters, of the sides of a
triangle are 24, 20, and 12. If the longest
sides of a similar triangle is 6 meters, what is
the length of the shortest side?
1. Draw a picture
12 m.
24 m.
3
x m.
6 m.
20 m.
2. Because they are similar, corresponding
sides are in proportion
24m.
12m.

6m.Aim: Similar Triangles
x
24x = (6)(12) = 72
x = 3 Course: Applied Geo.
PJ is 6-ft. tall. He casts
a shadow that is four
feet long. A nearby tree
of unknown height
casts a shadow of 30
feet. How tall is the
tree?
x
30 ft

4 x  6  30  180
6 ft
4 ft
Tree
Height
x
45
4 x 180

4
4
1
Aim: Similar Triangles
Tree’s shadow
30 ft.
x = 45 feet
PJ’s ht.
6 ft.
1 ~ 2
2
Course: Applied Geo.
PJ’s shadow - 4 ft.