Let P, Q, R be the mid-points of sides AB, BC, CA respectively of a triangle ABC

Geometry

Let P, Q, R be the mid-points of sides AB, BC, CA respectively of a triangle ABC. If the area of the triangle ABC is 5 square units, then the area of the triangle PQR is

  1. 5/3 square units
  2. 5/2√2 square units
  3. 5/4 square units
  4. 1 square unit

Answer

The line joining the midpoints of the sides of the triangle form four triangles, each of which is similar to the original triangle.

ΔABC ~ ΔPQR

In ΔABC, P and R are mid points of AB and AC respectively.

PR || BC (midpoint theorem)

In ΔABC and ΔAPR

∠A is common and ∠APR = ∠ABC (corresponding angles)

Therefore, ΔABC ~ ΔAPR (AA similarity)

In ΔABC and ΔPQR, since P, Q, R are the midpoints of AB, BC and AC respectively,

PR = ½BC; (midpoint theorem)

ΔABC ~ ΔPQR (SSS similarity)

Area(ΔPQR) / Area(ΔABC) = PR2/BC2 = (1/2)2 = 1/4

Area(ΔPQR) = Area(ΔABC)/4 = 5/4

The correct option is C.