In the figure below, $$a = 128$$ . Find x and y .
A) $x = 52$ , $y = 128$
B) $x = 128$ , $y = 128$
C) $x = 128$ , $y = 52$

Question

In the figure below, $$a = 128$$ . Find x and y . 
A) $x = 52$ , $y = 128$ 
B) $x = 128$ , $y = 128$ 
C) $x = 128$ , $y = 52$

Quizplus · Accepted Answer

By similar triangles, we have $\frac{y}{a}= \frac{y}{x} + \frac{a-y}{x+y}$. Substituting $a=128$, we get $\frac{y}{128} = \frac{y}{x} + \frac{128-y}{x+y}$. Rearranging and simplifying this gives, $x + y = 2y\left(\frac{x}{y}\right)$ or equivalently $2y-y = \frac{x}{y} \cdot y - \frac{x}{y} \cdot x$ or $y= \frac{x^2}{y}$, since $y>0$. Substituting $a=128$, we get $$\frac{128}{x}=\frac{x+y}{y} = \frac{x}{y} + 1 =\frac{x^2}{y^2}+1=\frac{x^2+y^2}{y^2}$$ or equivalently, $128y^2= x^2+y^2+xy^2$. Substituting $y^2=\frac{x^4}{y^2}$ from the previous equation to eliminate $y^2$ and then solving for $x$, we get $x=52$ and $y=128$. Therefore, the answer is $\boxed{\textbf{(A)}}$.

In the Figure Below a=128a = 128a=128 Find X and Y x=52x = 52x=52

In the Figure Below $a = 128$ Find X and Y $x = 52$