Triangle Medians — 3 Properties You Must Know

A median joins a vertex to the midpoint of the opposite side. These three facts appear again and again in CAT/XAT geometry. Learn the idea + a 10-second proof each.

Property 1 — Median to the hypotenuse

Statement. In a right triangle \(\triangle ABC\) with right angle at \(B\), let \(D\) be the midpoint of the hypotenuse \(AC\). Then

\[ BD=AD=CD=\frac{AC}{2}. \] Moreover, the circle with center \(D\) and radius \(\tfrac{AC}{2}\) passes through \(A,B,C\) (Thale’s circle).

Why. In a right triangle, the midpoint of the hypotenuse is equidistant from all three vertices (via congruent triangles / perpendicular-bisector logic).

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Median to hypotenuse equals half the hypotenuse — With \(\angle B=90^\circ\) and \(D\) midpoint of \(AC\), we get \(BD=AD=CD=AC/2\).

Property 2 — A right-angle converse from equal segments

Statement. In any triangle \(\triangle ABC\), if a point \(D\) on \(AC\) satisfies \(BD=AD=DC\), then \(\angle A=90^\circ\).

Proof in one line: From \(AD=BD\) we get isosceles \(\triangle ABD\Rightarrow \angle ABD=\angle ADB=x\). From \(BD=DC\) we get isosceles \(\triangle BDC\Rightarrow \angle CBD=\angle BCD=y\). Then \(\angle ABC=x+y\) and \(\angle ACB=y+x\). Hence \[ \angle A=180^\circ-(\angle ABC+\angle ACB) =180^\circ-2(x+y)=90^\circ. \]

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Equal segments imply right angle at A — Equal segments force complementary base angles; thus the included angle at \(A\) is right.

Property 3 — In a right triangle, that median is the shortest

In \(\triangle ABC\) right-angled at \(B\), let medians to sides \(a=BC\), \(b=CA\) (hypotenuse), \(c=AB\) be \(m_a, m_b, m_c\) respectively. Then

\[ m_b \lt m_a \quad \text{and} \quad m_b \lt m_c. \] Using Apollonius (median length): \[ m_a^2=\frac{2b^2+2c^2-a^2}{4},\qquad m_b^2=\frac{2c^2+2a^2-b^2}{4},\qquad m_c^2=\frac{2a^2+2b^2-c^2}{4}. \] With \(b^2=a^2+c^2\) (hypotenuse), \(m_b=\tfrac{b}{2}\). Moreover, \[ m_a^2-\Big(\tfrac{b}{2}\Big)^2=\frac{c^2}{4}>0,\qquad m_c^2-\Big(\tfrac{b}{2}\Big)^2=\frac{a^2}{4}>0, \] hence the strict inequalities.

Show figure

Shortest median in a right triangle — The median to the hypotenuse equals \(b/2\) and is smaller than the other two medians.

Quick Check

Q. In a right-angled triangle with hypotenuse \(AC=10\,\text{cm}\), find \(BD\) where \(D\) is the midpoint of \(AC\).

Show answer

By Property 1, \(BD=AC/2=5\,\text{cm}\).

Triangle Medians — 3 Properties

Triangle Medians — 3 Properties

Triangle Medians — 3 Properties You Must Know

Property 1 — Median to the hypotenuse

Property 2 — A right-angle converse from equal segments

Property 3 — In a right triangle, that median is the shortest

Quick Check

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