Rectangle ABCD is dilated to create rectangle A'B'C'D' . The width of rectangle ABCD is 12 feet. The width of rectangle A'B'C'D' is 8 feet. The area of rectangle ABCD is 72 square feet. What is the area of rectangle A'B'C'D' ?

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avrycarraway avrycarraway

05-11-2019
Mathematics

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Rectangle ABCD is dilated to create rectangle A'B'C'D' . The width of rectangle ABCD is 12 feet. The width of rectangle A'B'C'D' is 8 feet. The area of rectangle ABCD is 72 square feet. What is the area of rectangle A'B'C'D' ?

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Ashraf82 Ashraf82 · Answer 1 · 2019-11-19T12:21:26+01:00

The area of rectangle A'B'C'D' is 32 square feet

Step-by-step explanation:

A dilation is a transformation that produces an image that is the

same shape as the original, but is a different size

A dilation stretches or shrinks the original figure

The figure and its image after dilation are similar

If two rectangles are similar, then

[tex]\frac{w_{1}}{w_{2}}=\frac{l_{1}}{l_{2}}[/tex] = constant ratio
[tex]\frac{P_{1}}{P_{2}}=\frac{w_{1}}{w_{2}}[/tex]
[tex]\frac{A_{1}}{A_{2}}=(\frac{w_{1}}{w_{2}})^{2}[/tex]

∵ Rectangle ABCD is dilated to create rectangle A'B'C'D'

∴ Rectangle ABCD is similar to rectangle A'B'C'D'

∵ The width of rectangle ABCD is 12 feet

∵ The width of rectangle A'B'C'D' is 8 feet

∴ [tex]\frac{w_{1}}{w_{2}}=\frac{12}{8}[/tex]

- Simplify the ratio to its lowest term by divide up and down by 4

∴ [tex]\frac{w_{1}}{w_{2}}=\frac{3}{2}[/tex]

∴ The scale factor of dilation is [tex]\frac{3}{2}[/tex]

Let us use the rule of the similar rectangles above

∵ The area of rectangle ABCD is 72 square feet

∵ [tex]\frac{A_{1}}{A_{2}}=(\frac{w_{1}}{w_{2}})^{2}[/tex]

∴ [tex]\frac{72}{A_{2}}=(\frac{3}{2})^{2}[/tex]

∴ [tex]\frac{72}{A_{2}}=\frac{9}{4}[/tex]

- By using cross multiplication

∴ [tex]A_{2}[/tex] × 9 = 72 × 4

∴ 9 [tex]A_{2}[/tex] = 288

- Divide both sides by 9

∴ [tex]A_{2}[/tex] = 32 feet²

∴ The area of rectangle A'B'C'D' = 32 feet²

The area of rectangle A'B'C'D' is 32 square feet

Learn more:

You can learn more about area of shapes in brainly.com/question/6530759

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solvere solvere · Answer 2 · 2019-11-20T05:06:43+01:00

32 square feet

Further explanation

Given:

Rectangle ABCD is dilated to create rectangle A'B'C'D'.
The width of rectangle ABCD is 12 feet.
The width of rectangle A'B'C'D' is 8 feet.
The area of rectangle ABCD is 72 square feet.

Question:

What is the area of rectangle A'B'C'D' ?

The Process:

The Similarity Ratio

We calculate the similarity ratio of the two widths given.

Let the width of the rectangle ABCD as AD and the width of the rectangle A'B'C'D 'as A'D'.

[tex]\boxed{ \ The \ Similarity \ Ratio = k = \frac{A'D'}{AD} \ }[/tex]

[tex]\boxed{ \ k = \frac{8}{12} \ } \rightarrow \boxed{ \ k = \frac{2}{3} \ }[/tex]

Next to calculate the area of the rectangle A'B'C'D', let us do it in two ways. You can choose one of them.

First Way:

The relationship between the area of ABCD and the area of A'B'C'D' is as follows:

[tex]\boxed{ \ The \ area \ of \ A'B'C'D' = k^2 \times the \ area \ of \ ABCD \ }[/tex]

[tex]\boxed{ \ The \ area \ of \ A'B'C'D' = \big( \frac{2}{3} \big)^2 \times 72 \ square \ feet \ }[/tex]

[tex]\boxed{ \ The \ area \ of \ A'B'C'D' = \frac{4}{9} \times 72 \ square \ feet \ }[/tex]

Thus, the area of rectangle A'B'C'D' is 32 square feet.

Second Way:

Let us find out the length of ABCD, we just call it AB.

[tex]\boxed{ \ Length \times width = the \ area \ of \ rectangle \ ABCD \ }[/tex]

[tex]\boxed{ \ Length \times 12 \ feet = 72 \ square \ feet \ }[/tex]

[tex]\boxed{ \ Length = (72 \div 12) \ feet \ }[/tex]

[tex]\boxed{ \ Length \ of \ AB = 6 \ feet \ }[/tex]

Next, we find out the length of A'B'C'D', we just call it A'B'. Use the similarity ratio.

[tex]\boxed{ \ The \ Similarity \ Ratio = k = \frac{A'B'}{AB} \ }[/tex]

[tex]\boxed{ \ \frac{A'B'}{6} = \frac{2}{3} \ }[/tex]

Therefore, the length of A'B' is 4 feet.

And then, we calculate the area of A'B'C'D'.

[tex]\boxed{ \ The \ area \ of \ rectangle \ A'B'C'D' = length \times width \ }[/tex]

[tex]\boxed{ \ The \ area \ of \ rectangle \ A'B'C'D' = 4 \ feet \times 8 \ feet \ }[/tex]

Thus, the area of rectangle A'B'C'D' is 32 square feet.

Learn more

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Keywords: rectangle ABCD, dilated, to create, A'B'C'D', the width, 12 feet, the area, 72 square, 32, the similarity ratio, length, ways

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