CAT 2024 Slot 3 QA Question 16: The midpoints of sides AB, BC, and AC in $ riangle ABC$ are M, N, and P, respectively. The medians drawn from A, B, and C intersect the line segments MP, MN and NP at X, Y, and Z, respectively. If th

Question

The midpoints of sides AB, BC, and AC in $	riangle ABC$ are M, N, and P, respectively. The medians drawn from A, B, and C intersect the line segments MP, MN and NP at X, Y, and Z, respectively. If th

Accepted Answer

Use coordinates. Since area ratios are preserved under affine transformations, take a convenient triangle with $A = (0,0)$, $B = (2,0)$ and $C = (0,2)$.

$M = (1,0),\,N = (1,1),\,P = (0,1)$

Now find the three intersections.

Median from A is $AN: y = x$; line $MP: x + y = 1$
$X = (\dfrac{1}{2},\dfr