In the Cartesian coordinates plane, if points , , and represent the vertices of a right-angled triangle at , find the value of .
If the triangle is right-angled, it will obey Pythagoras' theorem. The right angle is at , so is the hypotenuse of the triangle.
For any 2 points and , the length of the straight line joining the points can be found using
So we use
Use this length formula to find expressions for , , and .
To find , we substitute our expressions for , , and into the Pythagoras equation given at the start of the solution, simplify and solve for .
Simplifying and rearranging gets us
The value of is .