ABC is an isosceles triangle such that AB = AC A has coordinates (4, 37) B and C lie on the line with equation 3y = 2x + 12 Find an equation of the line of symmetry of triangle ABC. Give your answer in the form px + qy = r where p, q and are integers (5

As we know that AC = AB, then it must be the case that BC is the base of the triangle (because it is isosceles). Therefore, the line of symmetry of the triangle ABC goes through the point A, and is perpendicular to the base BC. The equation of the line of the base BC is 3y = 2x +12, and so the gradient of the base is 2x/3. As the gradient of a perpendicular to a line is -1/m, the gradient of the line of symmetry must be -3x/2. Now we have the equation y = -3x/2 + c, and we also know that one of the points on this line is A, which has the coordinates (4,37). By inputting these values of x and y, c can be worked out to be 43. Therefore the equation of the line is y = -3x/2 +43.The question asks to give your answer in the form px + qy = r where p, q and are integers, and therefore we must multiply the above equation by 2 and rearrange, giving the answer 2y + 3x = 86.