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A B C is a right triangle right-angle...

`A B C` is a right triangle right-angled at `C` . Let `B C=a ,\ \ C A=b ,\ \ A B=c` and let `p` be the length of perpendicular from `C` on `A B` , prove that (i) `c p=a b` (ii) `1/(p^2)=1/(a^2)+1/(b^2)`

Text Solution

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(i) We have
`ar (Delta ABC)=(1)/(2)xx(1)/(2)xxABxxCD=(1)/(2)cp` [ taking BC as base]
`:. (1)/(2)cp=(1)/(2) ab rArr cp =ap`.

Hence, cp= ab.
`(ii) cp=ab rArr (1)/(p)=(c)/(ab)`
` :. (1)/p^(2)=(c^(2))/(a^(2)b^(2))=(b^(2)+a^(2))/(a^(2)b^(2)) " " [ :. AB^(2)=AC^(2)+BC^(2)]`
`=((b^(2))/(a^(2)b^(2))+(a^(2))/(a^(2)b^(2)))=((1)/(a^(2))+(1)/(b^(2)))`
Hnece, `(1)/(p^(2))=(1)/(a^(2)+(1)/(b^(2))`
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