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Question
If `a/b = c/d` Show that a + b : c + d = `sqrt(a^2 + b^2) : sqrt(c^2 + d^2)`.
Solution
Let `a/b = c/d = k`
⇒ a = bk and c = dk
L.H.S.
= `(a + b)/(c + d) = (bk + b)/(dk + d)`
= `(b(k + 1))/(d (k + 1)) = b/d`
R.H.S.
= `sqrt(a^2 + b^2)/sqrt(c^2 + d^2) = sqrt(b^2k^2 + b^2)/(sqrt(d^2k^2 + d^2)`
= `(b(sqrt(k^2 + 1)))/(d(sqrt(k^2 + 1))) = b/d`
L.H.S. = R.H.S.
Hence proved.
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