lesson 8.1 & 8.2 solving problems with ratio and proportion today, we will learn to… …find...
TRANSCRIPT
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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
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Ratio
A ratio is a comparison of two numbers written in
simplest form.
a a : b a to b b
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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
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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
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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
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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
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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
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Solve the proportion for x.
8. 2 8 7 x-2
2(x-2) = 56
2x - 4 = 56
2x = 60x = 30
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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
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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
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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
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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
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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
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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
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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?
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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 ?
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Geometric MeanThe geometric mean of two
positive numbers
(a and b) is …. a x
x b
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Find the geometric mean of the given numbers.
35 and 175
x ≈ 78.3
xx35
175=
x2 = 35(175)
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Lesson 8.3Similar Polygons
Today, we will learn to…
…identify similar polygons
...use similar polygons
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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
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B
C
DA
FG
HE
ABCD ~ EFGH
CDGH
ADEH
ABEF
BCFG
A E, B F, C G, D H
Statement of Proportionality
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Scale Factor
The scale factor is the ratio of the lengths of two corresponding sides.
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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
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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
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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?
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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
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5 =
3
6. RSTU ~ LMNO. Find the following.
125mT =mS =55
x 2.4
x
x = 4
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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
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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.
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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
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Theorem 8.1If 2 polygons are similar,
then the ratio of the perimeters is __________ the ratio of corresponding
side lengths.
equal to
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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
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Turn to page 145 in your workbook!
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Lesson 8.4
Proving Triangles are Similar Triangles
Today, we will learn to…
…identify similar triangles
...use similar triangles
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Postulate 25Angle-Angle (AA) Similarity
Two triangles are similar if 2 pairs of corresponding
angles are congruent.
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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
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Determine whether the triangles are similar. If they are, write a similarity statement.
2. G
H
JK
L 27˚
27˚
ΔGLH ~ Δ____G KJ
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4. If the triangles are similar, write a similarity statement.
31˚
47˚
not similar
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5. If the triangles are similar, write a similarity statement.
43˚
not similar
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6. The triangles are similar, find x.
3 5 7
x2=
3x = 10
x ≈ 3.33
yx2
53y2
3y = 14
y ≈ 4.67
73
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8. The triangles are similar. Find x. A B
C
D E
15
2518
9
x
15 925 x=
x = 15
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Are the triangles similar? If they are, write a similarity statement.
Not ~ XZW ~ XTY
T
YX
X
Z
W
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Are the triangles similar? If they are, write a similarity statement.
Not ~ABD ~ BCE
40
75
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Lesson 8.5
Proving Triangles are Similar Triangles
Today, we will learn to…
…use similarity theorems to prove that two triangles are similar
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Theorem 8.2Side-Side-Side (SSS)
Similarity
If all three corresponding sides are proportional, then
the triangles are similar.
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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
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Theorem 8.3Side-Angle-Side (SAS)
Similarity If two sides are proportional
and the angles between them are congruent, then
the triangles are similar.
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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
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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
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Separate the triangles if it helps.3. A
C D
106
ΔACD ~ ΔABE by
B E
A
53
SAS
3 5 6 10
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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
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x = 7.5
G
H
JK
L6 x
108
8 10 6 x
8x = 60
What can we
conclude?
Find x. GLH ~ GKJ
5.
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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.
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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
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Estimate the height of the tree.8.
4 ft.
6 ft.
16 ft.
x ft.
x = 24 feet
4 6
=16 x
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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
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Lesson 8.6Proportions and Similar Triangles
Today, we will learn to…
…use proportionality theorems to calculate segment lengths
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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
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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
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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
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Theorem 8.5Triangle Proportionality
Converse
If a line divides two sides of a triangle proportionally, then it is
parallel to the third side.
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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
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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
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Theorem 8.6
If three or more parallel lines intersect two transversals,
then they divide the transversals proportionally.
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9. Find x and y.
y24
x 10.5
78
x=
10.5
24 =
y
x = 12
y = 21
8 7
78
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10. Find x, y, and z.
x15
13 y
z10
30.4
15=
x 30.438
38x = 456
x = 12
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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
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Theorem 8.7An angle bisector of a triangle divides the opposite side into segments whose lengths are proportional to the other two
sides.
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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
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24
12
8
13. Find x.
?
8= 12
x 24-x
x = 9.624 - x
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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
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Lesson 8.7Dilations
Today, we will learn to…
…identify dilation
...use properties of dilations to create a perspective drawing
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Dilation
A dilation is a transformation that
results in a reduction or enlargement of a figure.
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=
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.
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C
3
6
Scale Factor = CP
CP
'
P
Q
R
P´
Q´
R´Reduction
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C
Scale Factor =
38
P
Q
R
P´
Q´
R´Reduction
3
8new image preimage
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C
2
5P
QR
P´
Q´
R´
Scale Factor =
5
2
CP
CP
'Enlargement
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C
P
QR
P´
Q´
R´5
15
Scale Factor =
Enlargement new image preimage
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C
P
Q
R
P´
Q´
R´
4
10
Find x.
x 6
x = 2.4 410
= x6
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2510
=
P
C
QR
P´
Q´
R´
10
25Find x.
10
4
5
y
x
x = 25x
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P
C
QR
P´
Q´
R´
10
25Find y.
10
4
5
y
x
y = 102510
=y4
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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…
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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?
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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)
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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
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Find x.
5126
=
12x = 30x = 2.5
x
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A’B’C’
(0, 6)(6, 6)
(4.5, 3)
A’ B’
C’
ABC
(0, 4)(4, 4)(3, 2)
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X’Y’Z’
(-0.75,-0.5)(2, 1)(1, -1)
XYZ
(-1.5, -1)(4, 2)(2, -2)
X’Y’
Z’