7.5 Proportions & Similar Triangles
Geometry
Mr. Peebles
Spring 2013
Geometry Bell Ringer
x
3
12 y
• Solve for x and y.
Small
Medium
Large
Leg Small Leg Large Hypotenuse
Geometry Bell Ringer
x
3
12 y
15
3 x
x
12
3 y
y
362 y452 x
6y53x
• Solve for x and y.
Small
Medium
Large
Leg Small Leg Large Hypotenuse
3 x
12 y
x 15
y
Daily Learning Target (DLT) Tuesday March 12, 2013
“I can understand, apply, and remember to use proportionality theorems to calculate segment lengths
such as determining the dimensions of a piece of land.”
Theorems 8.4 Triangle Proportionality Theorem
Q
S
R
T
U
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two side proportionally.
If TU ║ QS, then RT
TQ
RU
US =
Theorems 8.5 Converse of the Triangle Proportionality Theorem
Q
S
R
T
U
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
RT
TQ
RU
US = If , then TU ║ QS.
Ex. 1: Finding the length of a segment
In the diagram AB ║ ED, BD = 8, DC =
4, and AE = 12. What is the length of
EC?
128
4
C
B A
D E
Step:
DC EC
BD AE
4 EC
8 12
4(12)
8
6 = EC
Reason
Triangle Proportionality Thm.
Substitute
Multiply each side by 12.
Simplify.
128
4
C
B A
D E
=
=
= EC
So, the length of EC is 6.
Ex. 2: Determining Parallels
Given the diagram, determine whether MN ║ GH.
21
1648
56
L
G
H
M
N
Ex. 2: Determining Parallels
Given the diagram, determine whether MN ║ GH.
21
1648
56
L
G
H
M
N
LM
MG
56
21 =
8
3 =
LN
NH
48
16 =
3
1 =
8
3
3
1 ≠
MN is not parallel to GH.
Theorem 8.6
If three parallel lines intersect two transversals, then they divide the transversals proportionally.
If r ║ s and s║ t and l and m intersect, r, s, and t, then
UW
WY
VX
XZ =
m
s
Z
YW
XV
U
rt
Theorem 8.7
If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.
If CD bisects ACB, then AD
DB
CA
CB =
D
C
A
B
Ex. 3: Using Proportionality Theorems
In the diagram 1 2 3, and PQ = 9, QR = 15, and ST = 11. What is the length of TU?
11
15
9
3
2
1
S
T
UR
Q
P
SOLUTION: Because corresponding angles are congruent, the lines are parallel and you can use Theorem 8.6
PQ
QR
ST
TU =
9
15
11
TU =
9 ● TU = 15 ● 11 Cross Product property
15(11)
9
55
3 = TU =
Parallel lines divide transversals proportionally.
Substitute
Divide each side by 9 and simplify.
So, the length of TU is 55/3 or 18 1/3.
Ex. 4: Using the Proportionality Theorem
In the diagram, CAD DAB. Use the given side lengths to find the length of DC. 15
9
14
D
A
C
B
15
9
14
D
A
C
B
Solution:
Since AD is an angle bisector of CAB, you can apply Theorem 8.7. Let x = DC. Then BD = 14 – x.
AB
AC
BD
DC =
9
15
14-X
X =
Apply Thm. 8.7
Substitute.
Ex. 4 Continued . . .
9 ● x = 15 (14 – x)
9x = 210 – 15x
24x= 210
x= 8.75
Cross product property
Distributive Property
Add 15x to each side
Divide each side by 24.
So, the length of DC is 8.75 units.
Use proportionality Theorems in Real Life
Example 5: Finding the length of a segment
Building Construction: You are insulating your attic, as shown. The vertical 2 x 4 studs are evenly spaced. Explain why the diagonal cuts at the tops of the strips of insulation should have the same length.
Use proportionality Theorems in Real Life
Because the studs AD, BE and CF are each vertical, you know they are parallel to each other. Using Theorem 8.6, you can conclude that DE
EF
AB
BC =
Because the studs are evenly spaced, you know that DE = EF. So you can conclude that AB = BC, which means that the diagonal cuts at the tops of the strips have the same lengths.
Ex. 6: Finding Segment Lengths
In the diagram KL ║
MN. Find the values
of the variables.
y
x
37.5
13.5
9
7.5
J
M N
KL
Solution
To find the value of x, you can set up a proportion.
y
x
37.5
13.5
9
7.5
J
M N
KL
9
13.5
37.5 - x
x =
13.5(37.5 – x) = 9x
506.25 – 13.5x = 9x
506.25 = 22.5 x
22.5 = x
Write the proportion
Cross product property
Distributive property
Add 13.5x to each side.
Divide each side by 22.5
Since KL ║MN, ∆JKL ~ ∆JMN and JK
JM
KL
MN =
Solution
To find the value of y, you can set up a proportion.
y
x
37.5
13.5
9
7.5
J
M N
KL
9
13.5 + 9
7.5
y =
9y = 7.5(22.5)
y = 18.75
Write the proportion
Cross product property
Divide each side by 9.
7.5 Assignment
Pages 400-401 (1-24)
Geometry Exit Quiz – 10 Points
y
4
16 x
• Solve for x and y.
Small
Medium
Large
Leg Small Leg Large Hypotenuse
Top Related