Find the coordinates of a point in a given triangle

Question

I am trying to solve part iv of this question. The picture should explain the question. I know the answer is $8$,but have no clue how to arrive at this.

Answer 1

The point $(x_0,y_0)$ that lies in the middle of the 2 points $P$ and $Q$ is $$(x_0,y_0)=(4,4)$$ You are looking for an isosceles triangle, so that a line from $R$ to the base would be orthogonal to it.

Let the slope $m_1$ be the one corresponding to $QR$ and the slope $m_2$ be that of the middle point and $R$.

Demand $m_1\cdot m_2=-1$: $$m_1=\frac{6-0}{8-2}=\frac{1}{2}$$and $$m_2=\frac{4-r}{4-2} $$and so you can solve for $r$:$$\frac{1}{2}\cdot\frac{4-r}{4-2}=-1$$ to get $$r=8$$

Answer 2

$$RQ^2=RP^2\to (2-0)^2+(2-r)^2=(8-2)^2+(6-r)^2\\ 8-4r=72-12r\to r=8$$