If ad-bca-b-c-d-ac-bda-b-d-cthen show that each ratio is a - b - c - d 4

Let:ad bca b c+d=ac bda b d+c=kAccording #160;to #160;the #160;theorem #160;on #160;equal #160;ratios,ad bc+ac bda b c+d+a b d+c=k #8658;ad+c bd+c ...

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Byju's Answer

Standard IX

Mathematics

Three Points on a Line

If ad-bca-b-c...

Question

If $\frac{a d - b c}{a - b - c + d} = \frac{a c - b d}{a - b - d + c}$ then show that each ratio is $\frac{a + b + c + d}{4}$ .

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Solution

$L e t : \frac{a d - b c}{a - b - c + d} = \frac{a c - b d}{a - b - d + c} = k According to the theorem on e qual ratios, \frac{a d - b c + a c - b d}{a - b - c + d + a - b - d + c} = k \Rightarrow \frac{a (d + c) - b (d + c)}{2 (a - b)} = k \Rightarrow \frac{d + c}{2} = k . . . (i) A g a i n, a ccording to the theorem on e qual ratios, \frac{a d - b c - a c + b d}{a - b - c + d - a + b + d - c} = k \Rightarrow \frac{a (d - c) + b (d - c)}{2 (d - c)} = k \Rightarrow \frac{a + b}{2} = k . . . (ii) RHS : \frac{a + b + c + d}{4} = \frac{a + b}{4} + \frac{c + d}{4} From (i) & (ii), w e g e t : RHS : \frac{k}{2} + \frac{k}{2} = k ∴ \frac{a + b + c + d}{4} = k Hence, each ratio is equal to \frac{a + b + c + d}{4} .$

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If ad-bca-b-c+d=ac-bda-b-d+cthen show that each ratio is a+b+c+d4.

If $\frac{a d - b c}{a - b - c + d} = \frac{a c - b d}{a - b - d + c}$ then show that each ratio is $\frac{a + b + c + d}{4}$ .