Question

PQRS is a square, E and F are points on the sides QR and RS respectively. Show that $ar\left(△PQE\right)=ar\left(△PSF\right)$, when it is given that $QS||EF$.

Answer 1

We know that all four sides of a square are equal in length.

So $PQ=QR=RS=SP=a$

In $\Delta PQE$,

Area of triangle $=\frac{1}{2}\times base\times height$

Area of $\Delta PQE=\frac{1}{2}(PQ\times QE)$

Area of $\Delta PQE=\frac{1}{2}(a\times QE)$ ___(1)

In $\Delta PSF$,

Area of $\Delta PSF=\frac{1}{2}(PS\times SF)$

Area of $\Delta PSF=\frac{1}{2}(a\times SF)$

Here $SF=QE$

(because both are the distance between two parallel line as $SQ||EF$)

So,

Area of $\Delta PSF=\frac{1}{2}(Q\times QE)$ __(2)

From equation (1) & (2)

Area of $\Delta PQE=$ Area of $\Delta PSF$

$\therefore$ Hence proved.