Points P and Q lie on side AB and AC of triangle ABC respectively such that segment PQ is parallel to side BC. If the ratio of AP:PB is 1:4 and area of Δ APQ is 4 sq cm, what is the area of trapezium PQCB?

It is given that AP : PB = 1 : 4

Let AP = 1 cm and PB = 4 cm

Let area of trapezium PQCB = $$x$$ sq cm

In $$\triangle$$ APQ and $$\triangle$$ ABC

$$\angle$$ PAQ = $$\angle$$ BAC  (common)

$$\angle$$ APQ = $$\angle$$ ABC   (Alternate interior angles)

$$\angle$$ AQP = $$\angle$$ ACB    (Alternate interior angles)

=> $$\triangle$$ APQ $$\sim$$ $$\triangle$$ ABC

=> Ratio of Area of $$\triangle$$ APQ : Area of $$\triangle$$ ABC = Ratio of square of corresponding sides = $$(AP)^2$$ : $$(AB)^2$$

= $$\frac{(1)^2}{(1 + 4)^2} = \frac{4}{(4 + x)}$$

=> $$\frac{4}{4 + x} = \frac{1}{25}$$

=> $$4 + x = 25 \times 4 = 100$$

=> $$x = 100 - 4 = 96 cm^2$$

=> Ans - (C)