Question

BL and CM are medians of triangle ABC right angled at A prove that 4[BL^2+CM^2]=5 BC^2

Answer 1

To prove 4 [BL^2+CM^2] = 5BC^2

Since BL is the Medians
So AL = CL = 1/2 AC ....... (i)

Again Since CM is the Median
so AM = MB = 1/2 AB ....... (ii)
In triangle ABC,
BC^2 = AB^2 + AC^2 ........ (a)

In triangle BAL,
BL^2 = AB^2 + AL^2
BL^2 = AB^2 + (1/2 AC)^2 from (i)
On further solving we get
4BL^2 = 4AB^2 + AC^2 ........ (b)

Again in triangle MAC
CM^2 = AC^2 + AM^2
CM^2 = AC^2 + (1/2 AB)^2 from(ii)
On further solving we get
4CM^2 = 4AC^2 + AB^2 ........ (c)

Adding (b) and (c)
4BL^2+4CM^2 = 4AB^2 + AC^2 + 4AC^2 +AB^2
4 (BL^2 + CM^2) = 5AB^2 + 5AC^2
4 (BL^2 + CM^2) = 5 (AB^2 + AC^2)
4 (BL^2 + CM^2) = 5 BC^2 from (a)
Hence proved. It helps you

Answer 2

this is a NCERT questions... the trick is .... the line which touch by most other line should be used in half or twice manner