# Question #aa5c1

Mar 3, 2017

$144$ area units.

#### Explanation:

Let

$S = \left\mid \Delta A B C \right\mid$
${S}_{1} = \left\mid \Delta P G H \right\mid$
${S}_{2} = \left\mid \Delta P I D \right\mid$
${S}_{3} = \left\mid \Delta P F E \right\mid$

by similarity of triangles we have

${S}_{1} / S = {\left(\frac{\overline{H P}}{\overline{A B}}\right)}^{2}$
${S}_{2} / S = {\left(\frac{\overline{I D}}{\overline{A B}}\right)}^{2}$
${S}_{3} / S = {\left(\frac{\overline{P E}}{\overline{A B}}\right)}^{2}$

because in two similar triangles, the ratio of their areas is equal to the square of the ratio of their respective sides.

But $\overline{H P} = \overline{A I}$ and $\overline{P E} = \overline{D B}$

so

$\sqrt{{S}_{1} / S} + \sqrt{{S}_{2} / S} + \sqrt{{S}_{3} / S} = \left(\frac{\overline{A I}}{\overline{A B}}\right) + \left(\frac{\overline{I D}}{\overline{A B}}\right) + \left(\frac{\overline{D B}}{\overline{A B}}\right) = 1$

then

$S = {\left(\sqrt{{S}_{1}} + \sqrt{{S}_{2}} + \sqrt{{S}_{3}}\right)}^{2}$

or

$S = {\left(3 + 4 + 5\right)}^{2} = 144$