Question

ABC is a right triangle right angle at c.let BC=a,CA=b and AB=c and let p be the length of perpendicular from C on AB. prove that :
$\frac{1}{p ^{2} } = \frac{1}{ {a}^{2} } + \frac{1}{ {b}^{2} }$

Answer 1

Area of ΔABC = 1/2 × Base × Height

= 1/2 × BC × AC

= 1/2 ab

Area of ΔABC = 1/2 × Base × Height

= 1/2 × AB × CD

= 1/2 cp

⇒ 1/2 ab = 1/2 cp

⇒ ab = cp

Hence proved.

In right angled triangle ABC, AB2 = BC2 + AC2

c2 = a2 + b2

(ab/p)2 = a2 + b2

a2b2/p2 =a2+b2 -From proof (1) 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.

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