Question

In the figure, if ar (ΔRAS) = ar (ΔRBS) and ar (ΔQRB) = ar (ΔPAS) then show that both the quadrilaterals PQSR and RSBA are trapeziums.

Answer 1

ΔRAS = ΔRBS ………….. (1) 

Both the triangles are on the same base RS and between the same pair of lines RS and AB. 

As their areas are equal RS must be parallel to AB. 

⇒ RS//AB 

∴ □ABRS is a quadrilateral in which AB//RS.

∴ □ABRS (or) □RSBA is a trapezium. 

Now AQRB = APAS (given) 

⇒ ΔQRB – ΔRBS = ΔPAS – ΔRAS 

[from (1) ΔRBS = ΔRAS] 

⇒ ΔQRS = ΔPRS 

These two triangles are on the same base RS and between the same pair of lines RS and PQ. 

As these two triangles have same area RS must be parallel to PQ. 

⇒ RS // PQ 

Now in quad PQRS; PQ//RS. 

Hence □PQRS is a trapezium.