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Question
If a : b =c : d, then prove that `("a"^2 + "c"^2)/("b"^2 + "d"^2) = ("ac")/("bc")`
Solution
`"a"/"b" = "c"/"d" => "a" = "bc"/"d"`
To prove,
`("a"^2 + "c"^2)/("b"^2 + "d"^2) = ("ac")/("bd")`
LHS
`("a"^2 + "c"^2)/("b"^2 + "d"^2) `
`=(("bc"/"d")^2 + "c"^2)/("b"^2 + "d"^2)`
`= (("b"^2"c"^2)/"d"^2 + "c"^2)/("b"^2 + "d"^2)`
`= ("c"^2 ("b"^2 + "d"^2))/("d"^2("b"^2 + "d"^2))`
`= "c"^2/"d"^2`
RHS
`("ac")/("bd")`
`= (("bc")/"d" "c")/("bd")`
`="bc"^2/"bd"^2`
`= "c"^2/"d"^2`
LHS = RHS
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