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  
, 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  
  and we want to find the value of  

The gradients of AC and CB multiply to give  

The gradient of AC is  

The gradient of CB is  

\[\frac{k-5}{2} \frac{4-k}{3} =-1\]

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

This factorises to give  
\[k=7, \: 2\]
Obviously, from the diagram above  
  and the point C has coordinates  

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