CAT 2024 Slot 3 · Quant · Question 7

Joining the midpoints M, N and P forms the medial triangle of \triangle ABC.

Each side of the medial triangle is a midline, so it is half the corresponding side of \triangle ABC and parallel to it.

Halving every length quarters the area, so the area of \triangle MNP is \frac{1440}{4}=360 sq cm.

Now take the median from A. It runs to the midpoint of BC, which is exactly the point N.

Segment MP is the midline parallel to BC, so AN crosses it at the midpoint of MP — that point is X.

By the same reasoning the median from B runs to P and crosses the midline MN at its midpoint, which is Y.

Likewise the median from C runs to M and crosses the midline NP at its midpoint, which is Z.

So X, Y and Z are the midpoints of the sides of \triangle MNP.

That makes \triangle XYZ the medial triangle of \triangle MNP, and the same quartering applies again.

Therefore the area of \triangle XYZ is \frac{360}{4}=90 sq cm.

Shortcut. The construction is simply the medial triangle taken twice, and each pass divides the area by 4. So the answer is \frac{1440}{{4}^{2}}=\frac{1440}{16}=90, with no coordinates needed.