Question

PQRS is a square E is the mid point of QR and F is the mid point of RS .the ratio of the area of triangle PEF to the area of square PQRS is?

Answer 1

The ratio of the area of triangle PEF to the area of square PQRS will be 3 : 8

Step-by-step explanation:

See the attached diagram.

Now, from the right Δ PQE, applying Pythagoras Theorem,

PE² = PQ² + QE² = (2x)² + x² = 5x²

⇒ PE = x√5

Now, from the right Δ ERF, applying Pythagoras Theorem,

EF² = ER² + RF² = x² + x² = 2x²

⇒ EF = x√2

Similarly, from the right Δ PSF, applying Pythagoras Theorem,

PF² = PS² + SF² = (2x)² + x² = 5x²

⇒ PF = x√5.

So, the half perimeter of Δ PEF is = $S = \frac{x\sqrt{5} + x\sqrt{5} + x\sqrt{2}}{2} = x\sqrt{5} + \frac{x}{\sqrt{2}}$

So, the area of the triangle Δ PEF will be

= $\sqrt{s(s - x\sqrt{5})(s - x\sqrt{5})(s - x\sqrt{2})}$

= $\sqrt{(x\sqrt{5} + \frac{x}{\sqrt{2} })(\frac{x}{\sqrt{2}})(\frac{x}{\sqrt{2}}))(x\sqrt{5} + \frac{x}{\sqrt{2}} - x\sqrt{2})}$

= $\sqrt{2.943x (0.5x^{2} ) (1.5289x)}$

= 1.5x²

Now, area of PQRS = (2x)(2x) = 4x²

So, the ratio of the area of triangle PEF to the area of square PQRS will be 1.5x² : 4x² = 3 : 8 (Answer)