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 right-angled.
b) Show that the perpendicular bisectors of XZ and YZ intersect at the centre of the base.
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 (x-6)^2 + (y-10)^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.