Question

$P$ is a point inside the $\Delta ABC$ . Three straight lines parallel to $AB, \, BC$ and $AC$ are drawn which divides the triangle into six regions. If the area of three triangles formed in these regions are $9,16$ and $25$ square units, then the area (in sq. units) of $\Delta ABC$ is

• A. $81 \, $
• B. $256 \, $
• C. $144$
• D. $64$

Answer 1

Answer: (C)

$\because \Delta ABC, \, \Delta TUP, \, \Delta PDE, \, \Delta SPR$ all are similar
$\therefore $ $ratio \, of \, areas \, =\left(ratio \, of \, sides\right)^{2}$
Let, $PT=\alpha =AS$
$\therefore $ $\frac{S R}{P T}=\frac{4}{5}\Rightarrow SR=\frac{4 \alpha }{5}$
$\frac{P E}{P T}=\frac{3}{5}\Rightarrow PE=\frac{3 \alpha }{5}$
$\Rightarrow AC=AS+SR+RC$
$=\alpha +\frac{4 \alpha }{5}+\frac{3 \alpha }{5}$
$=\frac{12 \alpha }{5}$
$\therefore \frac{a r \left(\Delta A B C\right)}{a r \left(\Delta P D E\right)}=\frac{\left(A C\right)^{2}}{\left(P E\right)^{2}}=\frac{1 2^{2} \cdot \left(\alpha \right)^{2}}{5^{2} \cdot \left(\frac{3^{2} \left(\alpha \right)^{2}}{5^{2}}\right)}$
Hence, $ar\left(\Delta A B C\right)=16ar\left(\Delta P D E\right)=144 \, sq. \, units$