📐 Important Median Properties in Triangles

The median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Here are three important properties you must know for CAT/XAT level geometry:

🔸 Concept 1: Special Property of Median in Right Triangle

In a right-angled triangle, the median to the hypotenuse is half the length of the hypotenuse:

\[ \text{If } D \text{ is midpoint of } AC, \text{ then } BD = AD = DC = \frac{1}{2}AC \]

This implies that angle \( \angle A = 90^\circ \) when all three segments from triangle vertex to base and midpoints are equal.

🔸 Concept 2: Using Angles with Median

When \( BD = AD = DC \), angle at vertex A becomes \( 90^\circ \) because of triangle angle properties: \[ 2x + 2y = 180^\circ \Rightarrow x + y = 90^\circ \Rightarrow \angle A = 90^\circ \]

🔸 Concept 3: Median to Hypotenuse is the Shortest Median

In a right-angled triangle, the median to the hypotenuse is not only equal to half of the hypotenuse but also the smallest of all three medians.

\[ \text{In } \triangle ABC, \text{ if } \angle B = 90^\circ \text{ and } D \text{ is the midpoint of } AC,\text{ then } BD = \frac{1}{2}AC \]

💡 Final Note:

Recognizing when a median splits a triangle into two equal parts or leads to a right angle is a powerful shortcut in CAT/XAT problems. Always look for midpoint relationships and segment equalities!