Question

BL and CM are medians of a AABC right angled at A. If BL = 3, CM = 4, then value of BC is​

Answer 1

Refer to the attachment for figure:-

• To find this we have prove one thing.

We have to prove:-

• 4(BL²+CM²)=5BC²

↦Proof : -

REFER TO THE ATTACHMENT

BL and CM are medians of the Δ ABC in which ∠A = 90°

From Δ ABC ,

BC² = AB² + AC² (Pythagoras Theorem) [1]

From Δ ABL,

BL² = AL² + AB²

BL² = (AC/2)² + AB² (L is mid-point of AC)

BL² = AC²/4 + AB²

4 BL² = AC² + 4 AB² [2]

From Δ CMA,

CM² = AC² + AM²

CM² = AC² + (AB/2)² (M is mid-point of AB)

CM² = AC² + AB²/4

4 CM² = 4 AC² + AB² [3]

Adding [2] and [3] ,

4 (BL² + CM²) = 5 (AC² + AB²)

i.e. , 4 (BL² + CM²) = 5 BC² [From (1)]

Now:-

Put the values of BL and CM.

⇥4 (BL² + MC²) = 5 BC² (Proved)

⇥4 (3² + 4²) = 5BC²

⇥4( 9 + 16) =5 BC²

⇥4 × 25 = 5 BC²

⇥100 = 5 BC²

⇥20 = BC²

⇥BC = √20

⇥BC = 2√5

• Hence the length of BC is 2√5.

Hope it helps you ✔️