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If X/(B + C - A) = Y /(C + a - B) = Z/(A + B - C) , Then Prove that Each Ratio is Equal to the Ratio of (X + Y+Z)/(A + B + C) - ICSE Class 10 - Mathematics

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Question

If `"x"/("b + c - a") =" y" /("c + a - b") = "z"/("a + b - c")` , then prove that each ratio is equal to the ratio of `("x + y+z")/("a + b + c")`

Solution

`"x"/("b + c - a") =" y" /("c + a - b") = "z"/("a + b - c") = "k"`

x = k (b + c- a) 

Y = k (c+a-b)

z = k (a+b-c) 

Now ,

`("x + y+z")/("a + b + c")`

`= ("k"("b + c - a") + "k" ("c + a - b")+ "k" ("a + b - c"))/("a + b + c")`

`= ("k" ("b + c - a + c + a - b + a + b - c"))/("a + b + c")`

`= ("k"("a + b + c"))/("a + b + c") = "k"`

Hence , 

`"x"/("b + c - a") =" y" /("c + a - b") = "z"/("a + b - c") = ("x + y+z")/("a + b + c")`

Proved.

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APPEARS IN

 Frank Solution for Frank Class 10 Mathematics Part 2 (2016 to Current)
Chapter 9: Ratio and Proportion
Exercise 9.3 | Q: 11

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Solution If X/(B + C - A) = Y /(C + a - B) = Z/(A + B - C) , Then Prove that Each Ratio is Equal to the Ratio of (X + Y+Z)/(A + B + C) Concept: Ratio and Proportion Example.
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