Question

In ∆ABC, ∠BAC = 90°, seg BL and seg CM are medians of ∆ABC. Then prove that 4 (BL² + CM²) = 5 BC².

Given : ∠BAC = 90°

seg BL and seg CM are the medians.

To prove: 4(BL² + CM²) = 5BC²

Answer 1

Proof:

In ∆BAL, ∠BAL 90° [Given]

∴ BL² = AB² + AL² (i) [Pythagoras theorem]

In ∆CAM, ∠CAM = 90° [Given]

∴ CM² = AC² + AM² (ii) [Pythagoras theorem]

∴ BL² + CM² = AB² + AC² + AL² + AM² (iii) [Adding (i) and (ii)]

Now, AL = 1/2 AC and AM = 1/2 AB (iv) [seg BL and seg CM are the medians]

∴ BC² = AB² + AC² (vi) [Pythagoras theorem] 

∴ 4(BL² + CM²) = 5BC² [From (v) and (vi)]