Question

A(10, 0) and B( 5, 5) are two vertices of a triangle whose in centre is the origin. Find the coordinates of the third vertex.

Answer 1

See Fig.  Let C(h, k) be the third vertex. Since O(0, 0) is the incentre, BO and AO are the bisectors of the angles B and A, respectively. The equation of BO is

x - y = 0  ...(1)

The equation of AO is

y = 0    ...(2)

Equation of the side AB is

y = 0 + 5/10 + 5(x - 10)

x - 3y - 10 = 0    ....(3)

Since BO is the angle bisector of  √B, the image of C(h, k) in the line BO lies on the line AB. Since the equation of BO is y = x, the image of C(h, k) in BO is (k, h) and this lies on AB. Therefore

k - 3h = 10   ...(4)

The image of C(h, k) in the angle bisector AO lies on the side AB. That is, (h, k) lies on the side AB. Therefore, from Eq. (3), we get

h - (-k = 10)   ...(5)

Solving Eqs. (4) and (5), we get h = - 2 and k = 4. Hence, the third vertex is (-2, 4).