Question

the rectangle below is dilated by a scale factor of \\(\frac{1}{4}\\). find the perimeter and area of the rectangle below, as well as the perimeter and area of the dilated rectangle. express your answers as whole numbers, decimals, or fractions in simplest terms (no mixed numbers). figures are not necessarily drawn to scale. \
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answer attempt 1 out of 2 \
perimeter of given rectangle \\(\square\\) units \
perimeter of dilated rectangle \\(\square\\) units \
area of given rectangle \\(\square\\) units\\(^2\\) \
area of dilated rectangle \\(\square\\) units\\(^2\\) \
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submit answer

Explanation:

Step1: Perimeter of given rectangle

The formula for the perimeter of a rectangle is \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width. Here, \( l = 42 \) and \( w = 28 \).
\( P = 2(42 + 28) = 2\times70 = 140 \)

Step2: Area of given rectangle

The formula for the area of a rectangle is \( A = l\times w \).
\( A = 42\times28 = 1176 \)

Step3: Dimensions of dilated rectangle

The scale factor is \( \frac{1}{4} \). So the new length \( l' = 42\times\frac{1}{4} = \frac{21}{2} = 10.5 \) and new width \( w' = 28\times\frac{1}{4} = 7 \).

Step4: Perimeter of dilated rectangle

Using the perimeter formula for the dilated rectangle: \( P' = 2(l' + w') = 2(10.5 + 7) = 2\times17.5 = 35 \)

Step5: Area of dilated rectangle

Using the area formula for the dilated rectangle: \( A' = l'\times w' = 10.5\times7 = 73.5 \) (or as a fraction, \( \frac{21}{2}\times7=\frac{147}{2} = 73.5 \))

Answer:

Perimeter of given rectangle: \( 140 \) units
Perimeter of dilated rectangle: \( 35 \) units
Area of given rectangle: \( 1176 \) units²
Area of dilated rectangle: \( 73.5 \) (or \( \frac{147}{2} \)) units²