Question

Find the exact value and the approximate value of the area of the triangle

Answer 1

From the big triangle we know that:

$\begin{gathered} x^2+y^2=12^2 \\ x^2+y^2=144 \end{gathered}$

From the triangle on the right we also know that:

$h^2+9^2=y^2$

From the triangle on the left we know:

$3^2+h^2=x^2$

Adding the last two equations we have that:

$\begin{gathered} x^2+y^2=h^2+9^2+h^2+3^2 \\ x^2+y^2=2h^2+90 \end{gathered}$

Equating the last equation with the first one we have that:

$\begin{gathered} 2h^2+90=144 \\ 2h^2=144-90 \\ 2h^2=54 \\ h^2=\frac{54}{2} \\ h^2=27 \\ h=\sqrt[]{27} \end{gathered}$

Then, the height of the triangle is the squared root of 27.

Once we know the height we can calculate the area.

$\begin{gathered} A=\frac{1}{2}bh \\ =\frac{1}{2}12\sqrt[]{27} \\ =6\sqrt[]{27} \end{gathered}$

Therefore the exact value of the area is:

$6\sqrt[]{27}$

This can be approximated to (rounding on the hundreths):

$31.18$