Question

In $△ABC,\angle B={90}^o$. $\square PQRS$ is a square of side $5$ units and $\square WJYZ$ is a square of side $10$ units. $MS$ and $SY$ are the length and breadth respectively of rectangle $MNSY$. What is the length of $MS$ ?

• A. $10$ units
• B. $12$ units
• C. $15$ units
• D. $20$ units

Answer 1

The correct option is C $15$ units

Lets observe few points.

$\angle MPQ=\angle BMN$

$\angle PMQ=\angle BNM$ (because sum has to be ${180}^{\circ }$)

Also,

$\angle BNM=\angle NJW$

$\angle BMN=\angle WNJ$

Thus we can say,

$\angle MPQ=\angle WNJ$

$\angle PMQ=\angle NJW$

Finally, since all $3$ angles of $\triangle PQM= \triangle NWJ$

They are similar by AAA condition.

Let length $NW=x$,

then $MQ=x+5$

By ratio of sides of similar triangles,

$\frac{5}{x+5}$ = $\dfrac{x}{10}$

$x^2+5x-50=0$

$(x-5)(x+10)=0$

$x=5$ (neglect $x=-10$ as length can't be negative)

$MS=x+5+5=15$