Question

Find the area of the figure below. Use trigonometry and draw a diagram. No Pythagorean theorem.

Answer 1

The Solution.

Below is the given figure, which is a rhombus, since all the sides are equal.

For a rhombus, the diagonals intersect at a right angle, that is, 90 degrees.

Considering right-angled triangle DOC:

We shall find the length of the unknown diagonal by applying the Pythagorean Theorem.

$\begin{gathered} x^2+12^2=13^2 \\ x^2+144=169 \\ x^2=169-144=25 \\ \text{Taking the square root of both sides, we get} \\ x=\sqrt[]{25}=\pm5 \\ x=5\text{ (discard x=-5)} \end{gathered}$

So, the unknown diagonal is

$AC=2\times OC=2\times5=10$

Now, the area of the given figure can be calculated with the formula below:

$\text{Area =}\frac{1}{2}(BD\times AC)=\frac{1}{2}(24\times10)=24\times5=120\text{ square units}$

Hence, the correct answer is 120 square units.