Question

In an acute-angled triangle, express a median in terms of its sides.

Answer 1

We have,

In ΔABC, AD is a median.

Draw AE ⊥ BC

In ΔAEB, by pythagoras theorem

AB² = AE² + BE²

⇒ AB² = AD² − DE² + (BD − DE)²            [By Pythagoras theorem]

⇒ AB² = AD² − DE² + BD² + DE² − 2BD × DE

⇒ AB² = AD² + BD² − 2BD × DE

⇒ AB² = AD² + `"BC"^2/4` - BC x DE         ...(i)  [BC = 2BD given]

Again, In ΔAEC, by pythagoras theorem

AC² = AE² + EC²

⇒ AC² = AD² − DE² + (DE + CD)²             [By Pythagoras theorem]

⇒AC² = AD² + CD² + 2CD × DE

⇒ AC² = AD² + `"BC"^2/4` + BC x DE     ...(ii)   [BC = 2CD given]

Add equations (i) and (ii)

AB² + AC² = 2AD² + `"BC"^2/2`

⇒ 2AB² + 2AC² = 4AD² + BC²           [Multiply by 2]

⇒4AD² = 2AB² + 2AC² − BC²

⇒ AD² = `(2"AB"^2+2"AC"^2-"BC"^2)/4`