question_answer

ABC is a right angled triangle such that AB \[AB=\mathbf{x}-\mathbf{y},\mathbf{BC}=\mathbf{x}\] and \[\mathbf{CA}=\mathbf{x}-\mathbf{y}\], D is a point on BC such that \[BD=AB\]. The ratio of BD:DC for any value of x and y is given by

A)  3:2                              

B)  4:3             

C)  5:4                              

D)  3:1

Correct Answer: D

Solution :

(d): in right angle \[\Delta ABC\], By Pythagoras theorem \[{{(x+y)}^{2}}={{(x-y)}^{2}}+{{x}^{2}}\] \[\Rightarrow \] \[{{x}^{2}}+{{y}^{2}}+2xy={{x}^{2}}+{{y}^{2}}-2xy+{{x}^{2}}\] \[\Rightarrow \] \[4xy={{x}^{2}}\Rightarrow 4y=x\] Now, \[\frac{BD}{DC}=\frac{x-y}{y}=\frac{4y-y}{y}=\frac{3y}{y}=\frac{3}{1}\]