Question

In a Δ ABC, P and Q are respectively the mid-points of AB and BC and R is the mid-point of AP. Prove that:

(i) ar (Δ PBQ) = ar (Δ ARC)

(ii) ar (Δ PRQ) = $\frac{1}{2}$ ar (Δ ARC)

(iii) ar (Δ RQC) = $\frac{3}{8}$ ar (Δ ABC).

Answer 1

Given:

(1) In a triangle ABC, P is the mid-point of AB.

(2) Q is mid-point of BC.

(3) R is mid-point of AP.

To prove:

(a) Area of ΔPBQ = Area of ΔARC

(b) Area of ΔPRQ = Area of ΔARC

(c) Area of ΔRQC = Area of ΔABC

Proof: We know that each median of a triangle divides it into two triangles of equal area.

(a) Since CR is a median of ΔCAP

Therefore …… (1)

Also, CP is a median of ΔCAB.

Therefore …… (2)

From equation (1) and (2), we get

Therefore …… (3)

PQ is a median of ΔABQ

Therefore

Since

Put this value in the above equation we get

…… (4)

From equation (3) and (4), we get

Therefore …… (5)

(b)

…… (6)

…… (7)

From equation (6) and (7)

…… (8)

From equation (7) and (8)

(c)

= …… (9)