Question

In the given figure, PQRS is a ∥ gm. A straight line through P cuts SR at point T and QR produced at N. Prove that area of triangle QTR is equal to the area of triangle STN.

Answer 1

ΔPQT and parallelogram PQRS are on the same base PQ and between the same parallel lines PQ and SR.
∴ Δ(ΔPQT) = `(1)/(2)"A"`(parallelogram PQRS)  ....(i)

ΔPSN and parallelogram PQRS are on the same base PS and between the same parallel lines PS and QN.

∴ Δ(ΔPSN) = `(1)/(2)"A"`(parallelogram PQRS)  ....(ii)

Adding equations (i) and (ii), we get
∴ A(ΔPQT) + A(ΔPSN) = A(parallelogram PQRS)
⇒ A(quad. PSNQ) - A(ΔQTN) = A(parallelogram PQRS)
⇒ A(quad. PSNQ) - A(ΔQTN) = A(quad. PSNQ) - A(ΔSRN)
⇒ A(ΔQTN = A(ΔSRN)
Subtracting A(ΔRTN) from both the sides, we get
A(ΔQTN) - A(ΔRTN) = A(ΔSRn) - A(ΔRTN)
⇒ A(ΔQTR) = A(ΔSTN).