# If Y + Z a = Z + X B = X + Y C Then Show that X B + C − a = Y C + a − B = Z a + B − C . - Algebra

Sum

If    ( y + z)/a = ( z + x )/b = ( x + y )/ c  then show that x/[ b + c - a ] = y/[ c + a - b ] = z / ( a + b - c)

#### Solution

( y + z)/a = ( z + x )/b = ( x + y )/ c
By invertendo,
a/( y + z)= b /( z + x )= c/( x + y )

a/( y + z)= b /( z + x )= c/( x + y ) = ( b + c - a)/( z + x + x + y - y - z)                                         .......( Theorem of equal ratios)

⇒ a/( y + z)= b /( z + x )= c/( x + y ) = ( b + c - a)/[2x]    ......(1)

Now,

a/( y + z)= b /( z + x )= c/( x + y ) = [ c + a - b]/[ x + y +y +z - z -x ]                                              .......( Theorem of equal ratios)

⇒ a/( y + z)= b /( z + x )= c/( x + y ) = ( c + b - a)/[2y]    ......(2)
Also,
a/( y + z)= b /( z + x )= c/( x + y ) = [ a + b - c]/[ y + z + z + x - x - y ]                                              .......( Theorem of equal ratios)
⇒ a/( y + z)= b /( z + x )= c/( x + y ) = ( a + b - c)/[2z]    ......(3)
From (1), (2) and (3), we have
[ b + c - a]/(2x) = [ c + a - b]/( 2y) = ( a + b - c)/( 2z)
⇒ [ b + c -a ]/x = [ c + a - b]/y = ( a + b - c )/z
by invertendo,

⇒  x / [ b + c -a ]=  y / [ c + a - b]=  z / ( a + b - c )

Concept: Theorem on Equal Ratios
Is there an error in this question or solution?

#### APPEARS IN

Balbharati Mathematics 1 Algebra 9th Standard Maharashtra State Board
Chapter 4 Ratio and Proportion
Practice Set 4.4 | Q 3.4 | Page 73