Lesson 8.1 & 8.2
Solving Problems with Ratio and Proportion
Today, we will learn to…
…find and simplify ratios
...use proportions to solve problems
Ratio
A ratio is a comparison of two numbers written in
simplest form.
a a : b a to b b
Simplify the ratio.
K H D M D C M1. 2 m : 300 cm200 cm : 300 cm 2 : 3
2. 2 km : 600 m 2000 m : 600 m 10 : 3
3. 10 mm : 5.5 cm 10 mm : 55 mm 2 : 11
4. In the diagram, DE : EF is1 : 2 and DF = 45. Find DE and EF.
D E F
1x + 2x = 453x = 45x = 15
1 2
DE =EF =
x x
1530
5. In ΔABC, the measures of the angles are in the
extended ratio of 3:4:5. Find the measures of the angles.
12x = 180x = 15
3x + 4x + 5x = 180
°, °, °
What do we know about the angles of a triangle?
45 60 75
6. The perimeter of a rectangle is 70 cm. The ratio of the length to the width is 3 : 2. Find the length and the width of the rectangle.
3x
2x 2x
3x 3x+2x+3x+2x = 70
Length is Width is
10x = 70x = 7
2114
7. A triangle has an area of 48 m 2. The ratio of the base to the
height is 2 : 3. Find the base and height.
A = ½ bh48 = ½ (2x)(3x)48 = 3x2
16 = x2
4 = xbase is height is
8 m12 m
2x
3x
Solve the proportion for x.
8. 2 8 7 x-2
2(x-2) = 56
2x - 4 = 56
2x = 60x = 30
9. On a map, 2 inch = 180 miles. Two cities are about 2 ¾ inches apart.
Estimate the actual distance between them.
2 in 180 mi
2x = 180(2¾) x = 247.5 miles
2 ¾ inx mi
10. In a photograph taken from an airplane, a section of a city street is 3 1/2 inches long and
1/8 of an inch wide. If the actual street is 30 feet wide, how long is it?
1/8 30
x=
x = 840 feet
31/2
x = (3 )(30)1/8 1/2
11. AB : AC is 3 : 2. Find x.
3x+3
x+1 2=
3(x+1) = 2(x+3)3x+3 = 2x+6
x + 3 = 6x = 3
12. Given MN MP find PQ. NO PQ=
x
14-x
46
14-xx=
x = 8.4
4x = 6(14-x)
?
?
4x = 84 - 6x
4
M
6
N
O Q
P14
5
A
2B
C E
D
7+x
x
7
57
77+x
=
x = 2.8
Given AB AD find DE. AC AE=
13.
5(7+x) = 49
?
? 35+5x = 495x = 14
14. Standard paper sizes are all over the world. The sizes all have the same width-to-length ratios. Two sizes of paper shown are A4 and A3. Find x.
210 mm
x
x
420 mm
210 x
420x=
x2 = (210)(420)
x ≈ 297 mm
15. The batting average of a baseball player is the ratio of
the number of hits to the number of official at-bats.
x.308
643 1= x = (643)(.308)
x = 198 hits
In 1998, Sammy Sosa of the Chicago Cubs had 643 official at-bats and a batting average of .308. How many hits did Sammy Sosa get?
16. A wheelchair ramp should have a slope of 1/12. If a ramp rises 2
feet, what is its run?
1 2 ft
x12=
x = (12)(2 ft)
x = 24 feet
What is its length?
length2 = 22 + 242 length2 = 4 + 576
length2 = 580 length = 24.08 feet
2 ft ?
Geometric MeanThe geometric mean of two
positive numbers
(a and b) is …. a x
x b
Find the geometric mean of the given numbers.
35 and 175
x ≈ 78.3
xx35
175=
x2 = 35(175)
Lesson 8.3Similar Polygons
Today, we will learn to…
…identify similar polygons
...use similar polygons
Two polygons are similar ifall corresponding angles are congruent and corresponding
sides are proportional.
AB BC AC
ΔABC ~ Δ XYZ if
A B C X Y Z
XY YZ XZand
B
C
DA
FG
HE
ABCD ~ EFGH
CDGH
ADEH
ABEF
BCFG
A E, B F, C G, D H
Statement of Proportionality
Scale Factor
The scale factor is the ratio of the lengths of two corresponding sides.
6 8 10
1. Are the triangles similar? If they are, find the scale factor and write a
statement of similarity.
9 12 15
Yes, the scale factor is 2
3XAR ~ __ __ __
M N T
4.5 6 9
2. Are the triangles similar? If they are, find the scale factor and write a
statement of similarity.
6 8 12
Yes, the scale factor is 3
4LMN ~ __ __ __T P O
12 15 x
A
B
C
D
E
F10
12
1512 y
x
4. Δ ABC ~ Δ DEF
15 10
3 2
=
y 10 12
x 12 =
15 10
x = 1812 y =
15 10 y = 8
Scale Factor?
The triangles are similar. Find x and y.5.
AC
8 x
12
Map the triangles to find corresponding sides.
B
E
DF
9
y
18B
A C
x
12
8
B
A C
x
12
8
9 y 18 x 12 8
9 x =
x = 4
18 8
y 12 =
18 8
y = 27
5 =
3
6. RSTU ~ LMNO. Find the following.
125mT =mS =55
x 2.4
x
x = 4
7. You have a 3.5 inch by 5 inch photo that you want to enlarge. You want the enlargement to be 16 inches wide. How long will it be?
3.5 16x5
=
3.5x = (16)(5)x = 22.9 ≈ 23 inches
A triangular work of art and the frame around it are similar equilateral triangles.
12 in.
16 in.9. Find the ratio of the perimeters. (artwork : frame)
34
34
3648
8. Find the ratio of the artwork to the
frame.
The rectangles are similar.
11. Find the ratio of the perimeters.
45
45
2227.5
10. Find the ratio of corresponding sides.
47
5
8.75
220275
Theorem 8.1If 2 polygons are similar,
then the ratio of the perimeters is __________ the ratio of corresponding
side lengths.
equal to
12. The patio around a pool is similar to the pool. The perimeter of the pool is 96 feet. The ratio of the
patio to the pool is 3 to 2. Find the perimeter
of the patio.
3 x962
= 2x = (3)(96)
x = 144 feet
patiopool
Turn to page 145 in your workbook!
Lesson 8.4
Proving Triangles are Similar Triangles
Today, we will learn to…
…identify similar triangles
...use similar triangles
Postulate 25Angle-Angle (AA) Similarity
Two triangles are similar if 2 pairs of corresponding
angles are congruent.
Determine whether the triangles are similar. If they are, write a similarity statement.
1. R M
NL
27˚
LT S
35˚
80˚
65˚
80˚
ΔRTS ~ Δ____M
35˚65˚
LN
Determine whether the triangles are similar. If they are, write a similarity statement.
2. G
H
JK
L 27˚
27˚
ΔGLH ~ Δ____G KJ
4. If the triangles are similar, write a similarity statement.
31˚
47˚
not similar
5. If the triangles are similar, write a similarity statement.
43˚
not similar
6. The triangles are similar, find x.
3 5 7
x2=
3x = 10
x ≈ 3.33
yx2
53y2
3y = 14
y ≈ 4.67
73
8. The triangles are similar. Find x. A B
C
D E
15
2518
9
x
15 925 x=
x = 15
Are the triangles similar? If they are, write a similarity statement.
Not ~ XZW ~ XTY
T
YX
X
Z
W
Are the triangles similar? If they are, write a similarity statement.
Not ~ABD ~ BCE
40
75
Lesson 8.5
Proving Triangles are Similar Triangles
Today, we will learn to…
…use similarity theorems to prove that two triangles are similar
Theorem 8.2Side-Side-Side (SSS)
Similarity
If all three corresponding sides are proportional, then
the triangles are similar.
Determine whether the triangles are similar. If they are, write a
similarity statement.1. D
E
F8
10
12
A
B
C 1512
18
ΔACB ~ Δ____ by _____ DFE
12 15 188 10 12
scale factor?
3:2
SSS
Theorem 8.3Side-Angle-Side (SAS)
Similarity If two sides are proportional
and the angles between them are congruent, then
the triangles are similar.
Determine if the triangles are similar. If they are, write a
similarity statement.2. A
B
C
D
E
F
6
8
8
12
Not similar
8 12 6 8
Determine whether the triangles are similar. If they are, write a
similarity statement.3.
A
B
C D
E
5
5
3
3
ΔABE ~ Δ____ by _____ACD SAS
3 5 6 10
Scale Factor? 1:2
Separate the triangles if it helps.3. A
C D
106
ΔACD ~ ΔABE by
B E
A
53
SAS
3 5 6 10
Find x. GLH ~ GKJ4.
x = 7.5
G
H
JK
L6 x
108 10 + x14
8 10 14 10 + x14 10 + x
8(10 + x) = 140
x = 7.5
G
H
JK
L6 x
108
8 10 6 x
8x = 60
What can we
conclude?
Find x. GLH ~ GKJ
5.
Theorem 8.4Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other
two sides, then it divides the two sides proportionally.
Find x. The triangles are similar.
6. 7.
x12
14
39
x = 18 x = 4
12 x
5 10
x10
5
2
26 39
2 x
Estimate the height of the tree.8.
4 ft.
6 ft.
16 ft.
x ft.
x = 24 feet
4 6
=16 x
3 5.5
=
Estimate the height of the tree.9.
3 ft.
5.5 ft.
12 ft.
x ft.
x = 27.5 feet
15 x
Lesson 8.6Proportions and Similar Triangles
Today, we will learn to…
…use proportionality theorems to calculate segment lengths
Find the value of x.
1. 2.
x4
6
12
x = 4.8 x = 2.8 4 x
5 7
x7
5
2
10 12
2 x
Find the value of x.
3. 4. x5
7
15
x = 6.25 x ≈ 2.67
5 x
6 8
x8
6
2
2 x 12 15
Find the value of x.
5. 6.10
14 x
x = 21 x = 11
36 x
x 33
x20
10
24 14 10 30
3633
Theorem 8.5Triangle Proportionality
Converse
If a line divides two sides of a triangle proportionally, then it is
parallel to the third side.
Use similar triangles to find x.
7. 8.
12
7 8x
x = 6 x = 20
6 10
8 x
x
10
86
?7 ?8
16 12
12 x
Mid-Segment Theorem
The segment connecting the midpoints of two sides of a
triangle is parallel to the third side and is _____ as long.half
Theorem 8.6
If three or more parallel lines intersect two transversals,
then they divide the transversals proportionally.
9. Find x and y.
y24
x 10.5
78
x=
10.5
24 =
y
x = 12
y = 21
8 7
78
10. Find x, y, and z.
x15
13 y
z10
30.4
15=
x 30.438
38x = 456
x = 12
10. Find x, y, and z.
1215
13 y
z10
30.4
15=
12 y13
15y = 156
y = 10.4
15=
12 z10
15z = 120
z = 8x = 12
Theorem 8.7An angle bisector of a triangle divides the opposite side into segments whose lengths are proportional to the other two
sides.
11. Find x. 12. Find x.
21 =
24 x = 7
24
x 8
21
x 8
3
5
2x
3 =
2 x 5 x = 7.5
24
12
8
13. Find x.
?
8= 12
x 24-x
x = 9.624 - x
14. Find x and y.
18
16
8.5
What is another way to write y?
8.5 - x
18= 16
x 8.5-x
x = 4.5
8.5 - x = 4
y = 4
Lesson 8.7Dilations
Today, we will learn to…
…identify dilation
...use properties of dilations to create a perspective drawing
Dilation
A dilation is a transformation that
results in a reduction or enlargement of a figure.
=
1. A circle in a photocopier enlargement has a 6 inch diameter. If the enlargement percentage is 125%, what is the diameter of the preimage circle?
4.8 in.100 x
125 6 in.
C
3
6
Scale Factor = CP
CP
'
P
Q
R
P´
Q´
R´Reduction
C
Scale Factor =
38
P
Q
R
P´
Q´
R´Reduction
3
8new image preimage
C
2
5P
QR
P´
Q´
R´
Scale Factor =
5
2
CP
CP
'Enlargement
C
P
QR
P´
Q´
R´5
15
Scale Factor =
Enlargement new image preimage
C
P
Q
R
P´
Q´
R´
4
10
Find x.
x 6
x = 2.4 410
= x6
2510
=
P
C
QR
P´
Q´
R´
10
25Find x.
10
4
5
y
x
x = 25x
P
C
QR
P´
Q´
R´
10
25Find y.
10
4
5
y
x
y = 102510
=y4
Rectangle ABCD has vertices A (3,1) , B (3, 3) , C (2, 3), and D (2, 1). Find the coordinates of the dilation with center (0,0) and scale factor of 2.
Graph on next slide…
A (3,1) , B (3, 3) , C (2, 3), D (2, 1)
A’(6,2) , B’(6, 6) , C’(4, 6), D’(4, 2)
Scale Factor is 2
x
2x
A’
B’C’
D’
Do you notice a pattern?
Rectangle ABCD has vertices A (-3,3) , B (3, 6) , C (6, -3), and D (-3, -6). Find the coordinates of the dilation with center (0,0) and scale factor of 1/3.
(-1,1) B’
D’ C’ A’
(2, -1)
(1, 2)
(-1, -2)
A’ (-1,1) B’ (1,2) C’ (2,-1) D’(-1,-2)
Scale Factor is 1/3
A(-3,3) B(3, 6) C(6, -3) D(-3, -6)
A’B’
C’D’
A
B
C
D
Find x.
5126
=
12x = 30x = 2.5
x
A’B’C’
(0, 6)(6, 6)
(4.5, 3)
A’ B’
C’
ABC
(0, 4)(4, 4)(3, 2)
X’Y’Z’
(-0.75,-0.5)(2, 1)(1, -1)
XYZ
(-1.5, -1)(4, 2)(2, -2)
X’Y’
Z’