Calculating the length of $DP$ in a rectangle.

Question

$P$ is a point in rectangle $ABCD$. Calculate the length of $DP$ if $AP = 3$, $PC = 5$ and $BP = 4$.

How do I go about by doing this? I clearly can't use Pythagoras because the hypotenuse is not a straight line. I thought of using coordinate geometry.

Answer 1

Let $B=(0,0)$, $C=(c,0)$ and $A=(0,a)$. Let $P=(x,y)$. Then by the "distance formula" (essentially the Pythagorean Theorem)

$$x^2+y^2=16,$$

$$x^2+(y-a)^2=9,$$

$$(x-c)^2+y^2=25.$$

Add the last two, subtract the first. We get $$(x-c)^2+(y-a)^2=18.$$

Note that $D=(c,a)$.