Question

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Corresponding lengths in similar figures are given. Find the ratios (shaded to unshaded) of the perimeters and ares. Then find the unknown areas for each. ​

Answer 1

Question 32

• Scale Factor = 3/8
• Ratio of Perimeters = 8 : 3
• Ratio of Areas = 64 : 9
• Area of smaller figure = 144 mm²

Question 33

• Scale Factor = 10/7
• Ratio of Perimeters = 7 : 10
• Ratio of Areas = 49 : 100
• Area of smaller figure = 196 in²

Explanation:

Scale Factor: A number used as a multiplier when scaling an object.

Perimeter: The total distance around the outside of a 2D shape.

Area: The space occupied by a flat shape, measured in two dimensions.

Question 32

Scale Factor

To find the scale factor from the shaded figure to the unshaded figure, divide the given side length of the unshaded figure by the given side length of the shaded figure:

`\implies \sf (unshaded)/(shaded)=(9)/(24)=(3)/(8)`

Ratio of Perimeters

Perimeters are measured in one dimension (length). To find the ratio of the perimeters, divide the given side length of one figure by the given side length of the other:

`\sf \implies(shaded)/(unshaded)=(24)/(9)=(8)/(3)`

Therefore:

`\sf \implies shaded:unshaded=8:3`

Ratio of Areas

Area is measured in two dimensions. To find the ratio of the areas, divide the square of the given side length of one figure by the square of the given side length of the other:

`\sf \implies(shaded)/(unshaded)=(24^2)/(9^2)=(64)/(9)`

Therefore:

`\sf \implies shaded:unshaded=64:9`

Area of the smaller figure (unshaded)

Area of shaded figure = 1024 mm²

Let x = area of unshaded figure

Use the ratio found above:

`\begin{aligned}\implies \sf (shaded\:area)/(unshaded\:area) & =\sf (64)/(9)\\\\\sf (1024)/(x) & =\sf (64)/(9)\\\\\sf 1024 \cdot 9 & = \sf 64x\\\\\sf x & = \sf (1024 \cdot 9)/(64)\\\\\implies \sf x & = \sf 144\:\:mm^2\end{aligned}`

Question 33

Scale Factor

To find the scale factor from the shaded figure to the unshaded figure, divide the given side length of the unshaded figure by the given side length of the shaded figure:

`\implies \sf (ushaded)/(shaded)=(20)/(14)=(10)/(7)`

Ratio of Perimeters

Perimeters are measured in one dimension (length). To find the ratio of the perimeters, divide the given side length of one figure by the given side length of the other:

`\sf \implies(shaded)/(unshaded)=(14)/(20)=(7)/(10)`

Therefore:

`\sf \implies shaded:unshaded=7:10`

Ratio of Areas

Area is measured in two dimensions. To find the ratio of the areas, divide the square of the given side length of one figure by the square of the given side length of the other:

`\sf \implies(shaded)/(unshaded)=(14^2)/(20^2)=(49)/(100)`

Therefore:

`\sf \implies shaded:unshaded=49:100`

Area of the smaller figure (shaded)

Area of unshaded figure = 400 in²

Let x = area of shaded figure

Use the ratio found above:

`\begin{aligned}\implies \sf (shaded\:area)/(unshaded\:area) & =\sf (49)/(100)\\\\\sf (x)/(400) & =\sf (49)/(100)\\\\\sf x& = \sf (400 \cdot 49)/(100)\\\\\implies \sf x & = \sf 196\:\:in^2\end{aligned}`