A large square has side length 4 . It is divided into four identical trapezoids and a small square, as shown. The small square has side length 1 . What is the area of each trapezoid?
(A) $\frac{2}{3}$
(D) $\frac{15}{4}$
(B) 3
(C) $\frac{9}{2}$
(E) 15

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-033.jpg?height=349&width=317&top_left_y=869&top_left_x=1316)
Solution 1

The large square has side length 4 , and so has area $4^{2}=16$.

The small square has side length 1 , and so has area $1^{2}=1$.

The combined area of the four identical trapezoids is the difference in these areas, or $16-1=15$. Since the 4 trapezoids are identical, then they have equal areas, each equal to $\frac{15}{4}$.

Solution 2

Suppose that the height of each of the four trapezoids is $h$.

Since the side length of the outer square is 4 , then $h+1+h=4$ and so $h=\frac{3}{2}$.

Each of the four trapezoids has parallel bases of lengths 1 and 4, and height $\frac{3}{2}$.

Therefore, the area of each is $\frac{1}{2}(1+4)\left(\frac{3}{2}\right)=\frac{5}{2}\left(\frac{3}{2}\right)=\frac{15}{4}$.

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-066.jpg?height=363&width=396&top_left_y=1190&top_left_x=1580)

ANswer: (D)