Math Reasonings Solved Exercises

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?

  1. 4

  2. 8

  3. 12

  4. 2

  5. 6

Try to solve it before checking the answer.
  1. 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,
\[ \begin{aligned} A &= \frac{1}{2} (4) (4) – \frac{1}{2} (4) (2) \\[.5em] &= 8 – 4 \\[.5em] &= \boldsymbol{ 4 \, cm^2} \end{aligned} \]
\[ \begin{aligned} A = \frac{1}{2} (4) (4) – \frac{1}{2} (4) (2) = 8 – 4 = \boldsymbol{4} \end{aligned} \]

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