Exercise 13
The area of the square \(ABCD\) in the attached figure is 16. If \(O\) is the center of the square, what is the area of the shaded region?
-
4
-
8
-
12
-
2
-
6
Try to solve it before checking the answer.
- 4
Let \(A\) be the area of the shaded region. We have that:
\[
A = \text{area of } \triangle ABE \, – \, \text{area of } \triangle ABO
\]
If \(L\) is the side of the square, then \( L^2 = 16\); therefore, \( L = 4\).
The base of these triangles is \( \overline{AB} = 4\) and their heights measure 4 and 2, respectively. Then,
\[
\begin{aligned}
A &= \frac{1}{2} (4) (4) – \frac{1}{2} (4) (2)
\\[.5em]
&= 8 – 4
\\[.5em]
&= \boldsymbol{ 4 \, cm^2}
\end{aligned}
\]
\[
\begin{aligned}
A = \frac{1}{2} (4) (4) – \frac{1}{2} (4) (2)
= 8 – 4
= \boldsymbol{4}
\end{aligned}
\]