In triangle ABC, medians AD, BE, and CF intersect at the centroid G. What is the ratio of the area of triangle GAB to the area of triangle ABC?

Correct — B. Triangles GAB and CAB share the base AB, so their areas are in the ratio of their heights above AB.

Centroid split of median CF: CG : GF = 2 : 1
So GF = 1⁄3 of CF, with F on AB
Perpendiculars from G and C to AB scale the same way: G's height = 1⁄3 of C's

Area GAB : Area ABC = 1⁄3 : 1 = 1 : 3 → option (b)

Why the others are wrong

• (a)
  1:2 — 1 : 2 is triangle ABD (or ACD) to ABC, because median AD halves the triangle. G sits a third of the way up median CF from AB, so GAB is smaller than half.
• (c)
  2:3 — 2 : 3 is a length ratio, AG : AD on a median, or the area of GAB and GBC together. Triangle GAB alone is one of three equal parts.
• (d)
  1:4 — 1 : 4 is the ratio for triangle DEF, formed by joining the three midpoints. The centroid triangles GAB, GBC and GCA are larger, a third each.

A median halves a triangle's area: the two halves have equal bases and the same height.

The three medians go further and cut the triangle into six small triangles of equal area, each 1⁄6 of the whole. Triangle GAB is two of them, AGF and BGF, so it is 2⁄6 = 1⁄3 of ABC.

GBC and GCA are also 1⁄3 each, so the centroid divides the triangle into three equal parts.

Key facts

• The centroid divides each median in the ratio 2 : 1, the longer part towards the vertex.
• The three medians form six triangles of equal area.
• Each of GAB, GBC and GCA is 1⁄3 of triangle ABC.
• The triangle joining the three midpoints has 1⁄4 of the original area.

Common traps

• Taking G's height above AB as 2⁄3 of C's. The longer part CG lies towards the vertex, so G is only 1⁄3 of the way up from AB.
• Answering 1 : 2 from the median rule. That is triangle ABD, whose third vertex is D on BC, not G.

No directly related past PYQ was found.