## Finding a Coordinate of a Right ANgled Triangle Using Gradients

We can check if a triangle is right angled, given the coordinates of the vertices, by multiplying the gradients of the sides adjacent to the right angles together. If the result is
$-1$
, then the triangle is right angled, We can also use this condition to find the coordinates of a vertex. The triangle below has a right angled at
$C(4,k)$
and we want to find the value of
$k$
.

The gradients of AC and CB multiply to give
$-1$

$\frac{k-5}{4-2}=\frac{k-5}{2}$

$\frac{4-k}{7-4}=\frac{4-k}{3}$

Hence
$\frac{k-5}{2} \frac{4-k}{3} =-1$

Rearranging gives
$(k-5)(4-k) =-6 \rightarrow -k^2+9k-20=-6 \rightarrow k^2-9k+14=0$

This factorises to give
$(k-7)(k-2)=0$
.
Hence
$k=7, \: 2$
.
Obviously, from the diagram above
$k=2$
and the point C has coordinates
$(4,2)$