CAT 2022 Slot 3 · Quant · Question 7

Medians BD and CE run to the midpoints D of AC and E of AB, and they cross at the centroid O.

Joining E and D gives a midline of the triangle, so ED is parallel to BC and half its length.

That makes triangles ODE and OBC similar, with ratio 1:2, since ED is parallel to BC.

The centroid divides each median in the ratio 2:1 from the vertex, which matches that similarity.

Work with a convenient triangle to pin down the fraction: take A at the origin, B at \left( 1,0 \right) and C at \left( 0,1 \right), so the area is \frac{1}{2}.

Then D, the midpoint of AC, is \left( 0,\frac{1}{2}\right) and E, the midpoint of AB, is \left( \frac{1}{2},0 \right).

The centroid is the average of the vertices, \left( \frac{1}{3},\frac{1}{3}\right).

The area of triangle EOD from these coordinates works out to \frac{1}{24}.

Comparing with the whole triangle's \frac{1}{2}, the ratio is \frac{1/24}{1/2}=\frac{1}{12}.

Ratios of areas do not depend on the particular triangle chosen, so this holds in general.

Therefore the area of EOD is \frac{108}{12}=9 sq cm.

Shortcut. Triangle ODE is similar to OBC with ratio 1:2, so its area is a quarter of OBC's. And OBC is one-third of ABC, since the three triangles formed by joining the centroid to the vertices are equal. So EOD is \frac{1}{4}\times \frac{1}{3}=\frac{1}{12} of ABC.