Question

In the figure, IF is a the altitude intersecting side AR at F. Find FR. $\left[4\right]$

Answer 1

In triangle IAR, IF is the altitude so, it will be perpendicular the side AR.
$\therefore$ $\angle$IFR = 90°.
Now, from $△$IFA, using Pythagorean theorem:

${\mathrm{IA}}^2={\mathrm{IF}}^2+{\mathrm{AF}}^2\Rightarrow {25}^2={15}^2+{\mathrm{AF}}^2\Rightarrow {\mathrm{AF}}^2=625-225=400\Rightarrow \mathrm{AF}=20\mathrm{ft}$$\left[1.5\right]$

From $△$IFR, using Pythagorean theorem:

${\mathrm{IR}}^2={\mathrm{IF}}^2+{\mathrm{FR}}^2\Rightarrow {17}^2={15}^2+{\mathrm{FR}}^2\Rightarrow {\mathrm{FR}}^2=289-225=64\Rightarrow \mathrm{FR}=8\mathrm{ft}$$\left[1.5\right]$

Hence, $AR=AF+FR=20+8=28ft$$\left[1\right]$