Question

In triangleRST,angle S is right angle.The mid points of two sides RS ans ST are X and Y respectively.Let us prove that, RY^2+XT^2=5XY^2?

Answer 1

see explanation

Explanation:

!
Let `RX=XS=a, and SY=YT=b`, as shown in the figure.
`DeltaRSY, DeltaRST, DeltaXSY and DeltaXST` are right triangles.
By Pythagorean theorem, we get
`XY^2=a^2+b^2`
`RY^2=(2a)^2+b^2=4a^2+b^2`
`XT^2=a^2+(2b)^2=a^2+4b^2`
`=> RY^2+XT^2=4a^2+b^2+a^2+4b^2=5(a^2+b^2)=5XY^2`

Hence, `RY^2+XT^2=5XY^2` (proved).