A designer has a rectangular piece of fabric that measures 12 inches by 18 inches. The des… — SAT

A designer has a rectangular piece of fabric that measures $12$ inches by $18$ inches. The designer cuts out two identical right triangular pieces from the corners along one of the $12$ -inch sides. Each right triangular piece has a leg of length $4$ inches along the $12$ -inch side and a leg of length $x$ inches along the $18$ -inch side. If the area of the remaining piece of fabric is $180$ square inches, what is the value of $x$ ?

Answer: 9 inches

Answer

The correct answer is 9. The original area of the rectangular fabric is

12 \times 18 = 216

square inches. Two identical right triangles with leg lengths of

4

inches and

x

inches are cut out. The area of each triangle is

\frac{1}{2} \times 4 \times x = 2x

square inches. The total area of the two triangles is

2 \times 2x = 4x

square inches. Subtracting this from the original area gives the remaining area:

216 - 4x = 180

. Solving for

x

gives

4x = 36

, which simplifies to

x = 9

Step-by-Step Solution

Calculate the area of the original rectangular piece of fabric.

216 square inches

To find the initial area before any modifications are made, using the formula

\text{Area} = \text{length} \times \text{width}

Find the total area of the two cut-out right triangles in terms of

x

4x

square inches

Each right triangle has legs of

4

and

x

, so its area is

\frac{1}{2}(4)(x) = 2x

. The total area of two such identical triangles is

2(2x) = 4x

Set up an equation using the remaining area of the fabric.

216 - 4x = 180

The remaining area of

180

square inches is equal to the original area of

216

square inches minus the total area of the two cut-out triangles, which is

4x

Solve the equation for

x

x = 9

Isolating the variable

x

by subtracting

216

from both sides and then dividing by

-4

yields

x = 9

Key Concept

Area of composite shapes (rectangles and triangles)

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