Find the 3rd point in a right triangle

Question

I've got this question in my worksheet:

BAC is a right triangle in A, given B(3,-1) , C(1,3)

If length of AB = length of AC, find the coordinates of point A.

I tried everything possible to do, I've even graphed it on a paper as a right triangle, and according to my graph, length AC isn't equal to length of AB.

Please help.

Answer 1

HINT: $$\vec{AB}=(3-x;-1-y)$$ and $$\vec{AC}=(1-x;3-y)$$ and now calculate the dot-product

Answer 2

Step 1. Form equation of line perpendicular the line segment joining $B$ and $C$ and passing through midpoint of $BC$ i.e. $(2,1)$.It will be $2y-x=0$

(Here you get locus of points equidistant from $B$ and $C$)

Step 2. Take a parametric point $A$ on that line. i.e. $(2t,t)$

Step 3. Apply condition for $\text{BA} \bot \text{AC}$ :$$m_{AB} \cdot m_{AC} = -1 ~;~~~~~~~~\text{( $m$ denotes slope)} $$

You will get two points. On solving for $t$ : $$t = 0 , 2 \implies A(0,0) ~\text{or} ~A(4,2) $$