Post on 19-Jan-2016
Geometry
7.6 Proportional Lengths
Proportional Lengths
AC and XZ are divided proportionally if…
. . .
. . .X
A B C
Y Z
=BC
XYAB
YZ
Example:
2 9
184=
2
9 18
4
Triangle Proportionality Thm. If a line parallel to one side of a triangle intersects
the other two sides, then it divides those sides
proportionally. big A
small A
whole Abig B
small B
whole B=
side C1
side C2
=
=
= = = =
whole A
small A
whole B
small B
side C1
side C2
big A
small A
big B
small B
big A
whole A
big B
whole B
whole Bwhole A
big A big B
small A
small B
big A
big B
whole A
whole B
All of these proportions, and their inverses, work.The key is to use the easiest one to solve each problem.
Think of it as two separate similar triangles.
Corollary
If three // lines intersect two transversals,…then they divide the transversals
proportionally.
a
b
c
d =ab
cd
Triangle Angle Bisector Thm. If a ray bisects an angle of a triangle,… then it divides the opposite side into
segments proportional to the other two sides.a
bc
d
=ab
cd
Directions: Use the given information and the triangle above to find the missing segment.
3. GE = 15 DE = 27 HF = 20 DH = ?
4. DE = 20 HF = 4 DH = 12 DG = ?
E F
G H
D
Directions: Use the given information and the diagram to find x.
2. AB = 27 BC = 18 DE = x + 10 EF = x
3. AB = 25 – x BC = x DE = 16 EF = 4
A
B
C
D
E
F
Directions: is and angle bisector of . Find x.
2. RP = x PQ = 12.5 RS = 8 SQ = 10
3. RP = 10 PQ = 20 RS = x RQ = 15
P
QR S
PS .RPQ
HW
P. 271 (1-7) P. 272-273 (1-14, 20, 21) Quiz 7.4-7.6 Tomorrow
A few from the HW
P. 272 #5, #14
An Example
15
520
12
416
=
18 24
=
=
= =
= =
20
5
16
4
18
24
15
5
12
4
15
20
12
161620
15 12
5
4
15
12
20
16
Solve for x. (figure not to scale)
15
27
12
20
x
= x
15
12
20
= x
5
4
20
Reduce by 3
times 4 equals
times 4 equals
x = 16
Solve for x.
18
27
x
x + 10
=18
27
x
x + 10
Reduce by 9.
=2
3
x
x + 10
2(x + 10) = 3x
2x + 20 = 3x
x = 20
20
30
Solve for x.
1512
18
x
=12
15
x
18 - x
18 - x
Reduce by 3.
=4
5
x
18 - x
4(18 – x) = 5x
72 – 4x = 5x
9x = 72
x = 8
8 10