Posted by Manda on .
An army base is enclosed by a wire fence so that it forms a circular compound. The entrance to the base is located at X(3, 6) and the exit is at Y(9, 14). X and Y are end points of a diameter of the circle. A search tower is positioned at Z(2, 13) on the circumference of the circle.
a) Show that the triangle formed by XYZ is rightangled.
b) Show that the perpendicular bisectors of XZ and YZ intersect at the centre of the base.

geometry 
Steve,
the center of the circle is at O(6,10)
OX = 5
So, the equation of the circle with diameter XY and center O is (x6)^2 + (y10)^2 = 25
OZ = 5, so Z is on that circle. Any triangle with the circle's diameter as one side, and a point on the circle is a right triangle.
Since Z is on the circle, XZ and YZ are two chords; their perpendicular bisectors intersect at O.