6.1 ratios, proportions, and the geometric mean
TRANSCRIPT
6.1
ANSWER 288
1. Simplify .3x2
9x
2. Solve 2n = 18 • 32.
ANSWERx3
ANSWER 6
ANSWER 244. Solve x = √36 • 16.
3. Solve 4x = 2x + 12.
ANSWER 80º
5. A triangle has angle measures 18º and 82º. Find the measure of the third angle.
6.1 Ratios, Proportions, and the Geometric MeanBell Thinger
6.1
6.1Example 1
SOLUTION
64 m : 6 ma. b. 5 ft20 in.
b. To simplify a ratio with unlike units, multiply by a conversion factor.
a. Write 64 m : 6 m as . Then divide out the units and simplify.
64 m6 m
= 323
= 32 : 3
5 ft20 in. = 60
20 = 31
Simplify the ratio.
64 m6 m
= 5 ft20 in.
12 in.1 ft
.
6.1Guided Practice
Simplify the ratio.
1. 24 yards to 3 yards
ANSWER 8 : 1
2. 150 cm : 6 m
ANSWER 1 : 4
6.1Example 2
SOLUTION
Painting You are planning to paint a mural on a rectangular wall. You know that the perimeter of the wall is 484 feet and that the ratio of its length to its width is 9 : 2. Find the area of the wall.
Write expressions for the length and width. Because the ratio of length to width is 9 : 2, you can represent the length by 9x and the width by 2x.
STEP 1
6.1
STEP 2
Solve an equation to find x.
Formula for perimeter of rectangle
Substitute for l, w, and P.Multiply and combine like terms.
Divide each side by 22.
=2l + 2w P=2(9x) + 2(2x) 484= 48422x
Evaluate the expressions for the length and width. Substitute the value of x into each expression.
STEP 3
= 22x
Length = 9x = 9(22) = 198
Width = 2x = 2(22) = 44
Example 2
The wall is 198 feet long and 44 feet wide, so its area is198 ft 44 ft = 8712 ft .2.
6.1Example 3
Combine like terms.
SOLUTION
Triangle Sum Theorem
Divide each side by 6.= 30x
=6x 180
= 180°x° + 2x° + 3x°
ALGEBRA The measures of the angles in CDE are in the extended ratio of 1 : 2 : 3. Find the measures of the angles.
Begin by sketching the triangle. Then use the extended ratio of 1 : 2 : 3 to label the measures as x°, 2x°, and 3x°.
The angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.
6.1Guided Practice
3. The perimeter of a rectangular room is 48 feet and the ratio of its length to its width is 7 : 5. Find the length and width of the room.
ANSWER 14 ft, 10 ft
4. A triangle’s angle measures are in the extended ratio of 1 : 3 : 5. Find the measures of the angles.
ANSWER 20°, 60°, 100°
6.1
6.1Example 4
SOLUTION
a. 510
x16=
Multiply.
Divide each side by 10.
a. 510
x16=
= 10 x80
= x8
Write original proportion.
Cross Products Property
Solve the proportion.ALGEBRA
= 10 x5 16. .
6.1
SOLUTION
Solve the proportion.ALGEBRA
Subtract 2y from each side.
b. 1y + 1
23y=
= 2y + 23y
=y 2
Distributive Property
b. 1y + 1 = 2
3y
Write original proportion.
Cross Products Property
Example 4
= 2 (y + 1)1 3y.
6.1Guided Practice
5. 2 x
5 8=
6. 1x – 3
43x=
7.
y – 3 7
y14=
16 5
ANSWER
ANSWER 12
ANSWER 6
Solve the proportion.
6.1Example 5
SOLUTION
SCIENCE As part of an environmental study, you need to estimate the number of trees in a 150 acre area. You count 270 trees in a 2 acre area and you notice that the trees seem to be evenly distributed. Estimate the total number of trees.
Write and solve a proportion involving two ratios that compare the number of trees with the area of the land.
6.1Example 5
Simplify.
Cross Products Property
There are about 20,250 trees in the 150 acre area.
=2702
n150
n=20,250
number of treesarea in acres Write proportion.
=270 150 2 n. .
6.1
6.1Example 6
Simplify.
Definition of geometric mean
Substitute 24 for a and 48 for b.
Factor.
Find the geometric mean of 24 and 48.
SOLUTION
=x ab
= 224
The geometric mean of 24 and 48 is 24 ≈ 33.9.2
= 24 48.
= 24 24 2. .
6.1Guided Practice
Find the geometric mean of the two numbers.
9. 12 and 27
ANSWER 18
10. 18 and 54
ANSWER 18 3
11. 16 and 18
ANSWER 12 2
6.1Exit Slip
1. Simplify the ratio 1200 cm : 1.8 m.
ANSWER 20 : 3
2. The perimeter of a rectangle is 528 millimeters . The ratio of length of the width is 8 : 3. Find the length and the width.
ANSWER 192 mm, 72 mm
6.1Exit Slip
4. Find the geometric mean of 42 and 12.
ANSWER 6 14
3. Solves + 8 1 = 36
3
ANSWER 4
6.1Exit Slip
5. The extended ratio of the angles of a triangle is 5: 12: 13. Find the angle measures of the triangle.
30°, 72°, 78° ANSWER
6. The area of a rectangle is 720 square inches. If the ratio of the length to the width is 5 : 4, find the perimeter of the rectangle.
ANSWER 108 in.
6.1
Homework
Pg 376-379#12, 16, 28, 50, 58