The value of $k$ that makes $\overline{AC}+\overline{BC}$ as small as possible.

Question

Help with this problem please

The coordinates of $A,B$ and $C$ are $(5,5),(2,1)$ and $(0,k)$ respectively. The value of $k$ that makes $\overline{AC}+\overline{BC}$ as small as possible.

Answer 1

It is well known problem.

Reflect the $B$ across y-line to $B'(-2,1)$ and calculate where line $AB'$ cuts y-line. This is $D$.

Then by triangle inequality we have $$AC+BC = AC+B'C \geq AB'$$ So $AC+BC$ will be minimal exactly when $C=D$.

Answer 2

Consider point $D(-2,1)$ symmetric of $B$ wrt the $y-$axis. The segment $AD$ is the shortest distance between $A$ and $D$ and therefore, if $C$ is the intersection point of $AD$ with $y-$axis as $DC=BC$ the point $C$ is also such that $AC+BC$ is the minimum.

To know where the point $C$ exactly is write the equation of the line $AD$.

The slope is $\dfrac{5-1}{5+2}=\dfrac{4}{7}$ so we have

$y-1=\dfrac{4}{7}(x+2)$

The point $C$ is the point of the line such that $x=0$ so that

$y_C=k=\dfrac{8}{7}+1=\dfrac{15}{7}$