39 Let ABC be a right triangle and let AD be the altitude fr

3.9 Let ABC be a right triangle, and let AD be the altitude from the right angle A to the hypotenuse BC. Prove that AD2- BD x DC (in the sense of content).

Solution

In triangl ABD

AB^2=AD^2+BD^2 ------(1)

In triangle ADC

AC^2=AD^2+DC^2 -------(2)

Adding (1) and (2)

AB^2+AC^2= 2AD^2+BD^2+DC^2 -----(3)

since AB^2+AC^2=BC^2 because ABC is a right angle triangle hence (3) can be written as

BC^2= 2AD^2+ BD^2+DC^2

(BD+DC)^2= 2AD^2+BD^2+DC^2

BD^2+DC^2+2BD×DC= 2AD^2+BD^2+DC^2

2BD×DC=2AD^2

BD×DC= AD^2

Hence proved.