Length of a line

Geometry Level pending

In the figure shown above, square $ABCD$ has a side length of $8$ while square $DEFG$ has a side length of $6$ . $GE$ is extended to meet $BC$ at $H$ . $HG||CM$ . $K$ is a point on $AM$ such that, $HK \bot AM$ . Find the length of $HK$ . If your answer is of the form $a\sqrt{b}$ where $a$ and $b$ are positive coprime integers and $b$ is square-free, find $a+b$ .

Note:

$a$ and $b$ are also prime numbers.

The answer is 7.

2 solutions

Ahmad Saad
Jun 17, 2017

Marta Reece
Jun 17, 2017

The two red lines are parallel and offset from each other by $2$ in both the horizontal and vertical directions.

Line $AM$ has equation $y=\dfrac17x$

Line through $H$ perpendicular to it has equation $y=-7x+b$ with $b=50$

Intersection of these lines is where $\dfrac17x=-7x+50$ , which is at $x=7$ .

The corresponding $y=1$ , and the distance $\overline{HK}=\sqrt{50}=5\sqrt2$

The answer is $5+2=\boxed7$