A sign is in the shape of a parallelogram with a base of 18 inches and a height of 10 inch… — SAT

A sign is in the shape of a parallelogram with a base of $18$ inches and a height of $10$ inches. A smaller, similar parallelogram is cut out from the center of the sign. If the scale factor of the smaller parallelogram to the larger parallelogram is $1$ to $3$ , what is the area, in square inches, of the remaining portion of the sign?

Answer: 160 square inches

Answer

160

The area of the larger parallelogram is

18 \times 10 = 180

square inches. Since the smaller parallelogram is similar to the larger one with a scale factor of

1

3

, the area of the smaller parallelogram is

\left(\frac{1}{3}\right)^2 = \frac{1}{9}

of the area of the larger parallelogram. This gives an area of

180 \times \frac{1}{9} = 20

square inches for the smaller parallelogram. Subtracting this from the larger area yields

180 - 20 = 160

square inches.

Step-by-Step Solution

Calculate the area of the larger parallelogram.

A_{\text{larger}} = 180

square inches

The area of a parallelogram is the product of its base and height (

A = bh

Determine the area of the smaller, similar parallelogram.

A_{\text{smaller}} = 20

square inches

The ratio of the areas of similar figures is the square of the scale factor:

\left(\frac{1}{3}\right)^2 = \frac{1}{9}

Subtract the smaller area from the larger area to find the remaining area.

A_{\text{remaining}} = 160

square inches

The remaining portion of the sign is the total area minus the cutout area.

Key Concept

The area of similar geometric shapes scales by the square of the linear scale factor.

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