Amiya Kumar 
📐 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!
