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Question
If a, b, c, d are in proportion, prove that (a2 + c2) : (b2 + d2) = ac : bd
Theorem
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Solution
`a/b = c/d` = k
a = bk and c = dk
L.H.S.
= `(a^2 + c^2)/(b^2 + d^2)`
= `((bk)^2 + (dk)^2)/(b^2 + d^2)`
= `(b^2k^2 + d^2k^2)/(b^2 + d^2)`
= `(k^2(b^2 + d^2))/(b^2 + d^2)`
= k2
R.H.S.
= `(ac)/(bd)`
= `((bk)(dk))/(bd)`
= `(bdk^2)/(bd)`
= `(bdk^2)/(bd)`
= k2
L.H.S. = R.H.S.
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Chapter 7: Ratio and proportion - Exercise 7B [Page 126]
