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Mathematics

If ab=cd\dfrac{a}{b} = \dfrac{c}{d}, show that :

(a + b) : (c + d) = a2+b2:c2+d2\sqrt{a^2 + b^2} : \sqrt{c^2 + d^2}

Ratio Proportion

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Answer

Let ab=cd\dfrac{a}{b} = \dfrac{c}{d} = k,

a = bk, c = dk.

Substituting a = bk, c = dk in L.H.S. of (a + b) : (c + d) = a2+b2:c2+d2\sqrt{a^2 + b^2} : \sqrt{c^2 + d^2}

L.H.S.=a+bc+d=bk+bdk+d=b(k+1)d(k+1)=bd.\text{L.H.S.} = \dfrac{a + b}{c + d} \\[1em] = \dfrac{bk + b}{dk + d} \\[1em] = \dfrac{b(k + 1)}{d(k + 1)} \\[1em] = \dfrac{b}{d}.

Substituting a = bk, c = dk in R.H.S. of (a + b) : (c + d) = a2+b2:c2+d2\sqrt{a^2 + b^2} : \sqrt{c^2 + d^2}

R.H.S.=a2+b2c2+d2=(bk)2+b2(dk)2+d2=b2(k2+1)d2(k2+1)=b2d2=bd.\text{R.H.S.} = \dfrac{\sqrt{a^2 + b^2}}{\sqrt{c^2 + d^2}} \\[1em] = \dfrac{\sqrt{(bk)^2 + b^2}}{\sqrt{(dk)^2 + d^2}} \\[1em] = \dfrac{\sqrt{b^2(k^2 + 1)}}{\sqrt{d^2(k^2 + 1)}} \\[1em] = \dfrac{\sqrt{b^2}}{\sqrt{d^2}} \\[1em] = \dfrac{b}{d}.

Since, L.H.S. = R.H.S. = bd\dfrac{b}{d}

Hence, proved that (a + b) : (c + d) = a2+b2:c2+d2\sqrt{a^2 + b^2} : \sqrt{c^2 + d^2}.

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