In a triangle ABC, BD and CE are two medians perpendicular to ea

| In a triangle ABC, BD and CE are two medians perpendicular to each other. If AB = 6 cm and AC = 13 cm, find BC?

A. √41 cm

B. 6√5 cm

C. 7 cm

D. √39 cm

Please scroll down to see the correct answer and solution guide.

Right Answer is: A

SOLUTION

Method 1:

We know that, in such case
AB² + AC²= 5BC²6² +13²= 5BC²
BC²= 205/5=41
BC= √41

Method 2:
According to the question;
∠EOB = ∠COD = ∠BOC = ∠EOD = 90°
BE = AB/2; CD= AC/2
in ΔBOE,
BE² = OB²+OE²AB²/4 = OB²+OE²
AB² = 4OB²+4OE²........(i)
in ΔCOD,
DC² = OC²+OD²AC²/4 = OC²+OD²AC² = 4OC²+4OD².........(ii)
in ΔBOC,
BC² = OB²+OC² ........(iii)
Adding (i) and (ii)
AB²+AC²=4OB²+4OE²+4OC²+4OD²
AB²+AC²=4BC²+OB²+OC²AB²+AC²=5BC²
6² +13²= 5BC²
BC²= 205/5=41
BC= √41