7.5 Proportions In Triangles Objective: Objective: Use Side-Splitter theorem & Triangle-Angle-...
-
Upload
anna-mills -
Category
Documents
-
view
216 -
download
0
Transcript of 7.5 Proportions In Triangles Objective: Objective: Use Side-Splitter theorem & Triangle-Angle-...
7.5 Proportions In 7.5 Proportions In TrianglesTriangles
Objective:Objective: Use Side-Splitter theorem & Triangle-Angle-Bisector Theorem to calculate segment lengths.
Theorems 7.4 Side-Splitter 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
RUUS
=
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
ReasonSide-Splitter Thm.
SubstituteMultiply each side
by 12.Simplify.
128
4
C
B A
D E
=
=
= EC
Ex. 2: Determining Parallels
Given the diagram, determine whether MN ║ GH.
21
1648
56
L
G
H
M
N
LMMG
56
21=
8
3=
LN
NH
48
16=
3
1=
8
3
3
1≠
MN is not parallel to GH.
Corollary to Theorem 7-4 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 U
WWY
VX
XZ=
l
m
s
Z
YW
XV
U
rt
Triangle-Angle-Bisector Thm
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 Triangle-Angle-Bisector Theorem
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
PQ
QR
ST
TU=
9
15
11
TU=
9 ● TU = 15 ● 11 Cross Multiply
15(11)
9
55
3=TU =
Parallel lines divide transversals proportionally.
Ex. 4: Using the Proportionality Theorem
In the diagram, CAD DAB. Use the given side lengths to find the length of DC.
15
9
14D
A
C
B
15
9
14D
A
C
B
Solution:
Since AD is an angle bisector of CAB, you can apply Theorem 7.5. Let x = DC. Then BD = 14 – x.
AB
AC
BDDC
=
9
15
14-X
X=
Ex. 4 Continued . . .
9 ● x = 15 (14 – x)
9x = 210 – 15x
24x= 210
x= 8.75
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
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
Homework
Page 401# 8 – 26 All