Question

The medians QM and RN of ΔPQR intersect at O. Prove that: area of ΔROQ = area of quadrilateral PMON.

Answer 1

Join MN. Since the line segment joining the mid-points of two sides of a triangle is parallel to the third side; so, MN || QR
Clearly, ΔQMN and ΔRNM are on the same base MN and between the same parallel lines.
Therefore, area(ΔQMN) = area(ΔRNM)
⇒ Area(ΔQMN) - area(ΔONM) = area(ΔRNM) - area(ΔONM)
⇒ Area)ΔQON) = area (ΔROM)      ......(i)
We know that a median of a triangle divides it into two triangles of equal areas.
Therefore, area(ΔQMR) = area(ΔPQM)
⇒ area(ΔROQ) + area(ΔROM) = area(quad, PMON) + area(ΔQON)
⇒ area(ΔROQ) + area(ΔROM) = area(quad. PMON) + area(ΔROM) ...(from (i))
⇒ area(ΔROQ) = area(quad. PMON).