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A rectangle with an area of 144 units
^(2) is the image of a rectangle that was dilated by a scale factor of
(1)/(3). Find the area of the preimage, the original rectangle, before its dilation. Round your answer to the nearest tenth, if necessary.
Answer: units
^(2)

A rectangle with an area of $144$ units $^{2}$ is the image of a rectangle that was dilated by a scale factor of $\frac{1}{3}$ . Find the area of the preimage, the original rectangle, before its dilation. Round your answer to the nearest tenth, if necessary. $\newline$ Answer: units $^{2}$

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Dilations and parallel lines

Full solution

Q. A rectangle with an area of

144

units

^{2}

is the image of a rectangle that was dilated by a scale factor of

\frac{1}{3}

. Find the area of the preimage, the original rectangle, before its dilation. Round your answer to the nearest tenth, if necessary.

\newline

Answer: units

^{2}

Understand Area Calculation: To find the area of the original rectangle before dilation, we need to understand that the area of a figure after dilation is the square of the scale factor times the area of the original figure. In this case, the scale factor is $\frac{1}{3}$ , so we need to find the area of the original figure by dividing the area of the dilated figure by the square of the scale factor. $\newline$ Calculation: Area of original rectangle = Area of dilated rectangle / (scale factor) $^2$ $\newline$ Area of original rectangle = $144$ units $^2$ / $(\frac{1}{3})^2$
Calculate Scale Factor Square: Now we calculate the square of the scale factor, which is $(\frac{1}{3})^2$ . $\newline$ Calculation: $(\frac{1}{3})^2 = \frac{1}{9}$
Divide by Scale Factor Square: Next, we divide the area of the dilated rectangle by the square of the scale factor to find the area of the original rectangle. $\newline$ Calculation: Area of original rectangle = $144 \, \text{units}^2 / (1/9)$
Perform Division: Perform the division to find the area of the original rectangle. $\newline$ Calculation: Area of original rectangle = $144 \, \text{units}^2 \times \frac{9}{1} = 1296 \, \text{units}^2$

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