Question

In the figure above, point B lies on. $\overline{AC}$. If x and y are integers, which of the following is a possible value of x ?

• A. 30
• B. 35
• C. 40
• D. 50
• E. 55

Answer 1

The correct option is A 30

From the given figure,

Sum of the angles in a straight line $=$ $180$

$x$ $+$ $y$ $+$ $y$ $+$ $y$ $=$ $180$

$x$ $+$ $3y$ $=$ $180$

$3y$ $=$ $180$ $-$ $x$

$3$ $\times$ $y$ $=$ $180$ $-$ $x$

$y$ $=$ $\frac{180-x}{3}$

To find a possible value of ${}^{\prime }{x}^{\prime }$,

Check the given options for a possible value

Given that ${}^{\prime }{x}^{\prime }$ and ${}^{\prime }{y}^{\prime }$ are integers implies $x$ and $y$ can only take integer values,

$\left(A\right)$ When $x$ $=$ $30$,

$\Rightarrow y$ $=$ $\frac{180-x}{3}$

$\Rightarrow y$ $=$ $\frac{180-30}{3}$

$\Rightarrow y$ $=$ $\frac{150}{3}$

$\Rightarrow y$ $=$ $50$

$\left(B\right)$ When $x$ $=$ $35$,

$\Rightarrow y$ $=$ $\frac{180-x}{3}$

$\Rightarrow y$ $=$ $\frac{180-35}{3}$

$\Rightarrow y$ $=$ $\frac{145}{3}$

$\Rightarrow y$ $=$ $48.3$

$\left(C\right)$ When $x$ $=$ $40$,

$\Rightarrow y$ $=$ $\frac{180-x}{3}$

$\Rightarrow y$ $=$ $\frac{180-40}{3}$

$\Rightarrow y$ $=$ $\frac{140}{3}$

$\Rightarrow y$ $=$ $46.7$

$\left(D\right)$ When $x$ $=$ $50$,

$\Rightarrow y$ $=$ $\frac{180-x}{3}$

$\Rightarrow y$ $=$ $\frac{180-50}{3}$

$\Rightarrow y$ $=$ $\frac{130}{3}$

$\Rightarrow y$ $=$ $43.3$

$\left(E\right)$ When $x$ $=$ $55$,

$\Rightarrow y$ $=$ $\frac{180-x}{3}$

$\Rightarrow y$ $=$ $\frac{180-55}{3}$

$\Rightarrow y$ $=$ $\frac{125}{3}$

$\Rightarrow y$ $=$ $41.7$

${}^{\prime }{x}^{\prime }$ and ${}^{\prime }y$ are integers only when $x$ $=$ $30$ among these options.

Therefore, option $\left(A\right)$ is correct.