ABC is a triangle right angled at C. If p is the length of perpendicular from C to AB and . . .

Question :  ABC is a triangle right angled at C. If p is the length of perpendicular from C to AB and AB=c, BC=a and CA=b, then prove that
(i) pc=ab (ii) 1/p²=1/a²+1/b²

Doubt by Muskan

Solution :

i) We know, 
Area of Triangle = ½×Base×Height
ar(∆ABC) = ½ba — (1)
Also, ar(∆ABC) = ½cp — (2)
From (1) & (2)

½ba=½cp
ba=cp
pc=ab
c=ab/p — (3)

ii) In Rt. ∆ ABC
c²=a²+b² (By Pythagoras Theorem

(ab/p)²=a²+b² [Using eq (3)]
a²b²/p² = a²+b²
1/p² = (a²+b²)/a²b²
1/p² = (a²/a²b²) + (b²/a²b²)
1/p² = 1/b²+1/a²
1/p² = 1/a²+1/b²

Hence Proved