Finding the Coordinates of a Triangle Vertex
Date: 10/26/1999 at 12:49:37 From: madhusudhan Subject: Analytical geometry problem Respected sir, I am unable to solve this problem. Please help me out. The problem is: The points B and C of triangle ABC lie on the line 3y = 4x and y = 0 respectively. The line BC passes through (2/3,2/3). If O is the origin and AOBC forms a rhombus, find the co-ordinates of A. I tried to solve this problem by drawing a diagram, but I could not get any ideas. Please help me out.
Date: 10/26/1999 at 14:36:03 From: Doctor Rob Subject: Re: Analytical geometry problem Thanks for writing to Ask Dr. Math, Madhusudhan. The coordinates of B must be (3*a,4*a) for some a, because B lies on 3*y = 4*x. The coordinates of C must be (b,0) for some b, because C lies on y = 0. The line BC has equation (y-0)*(3*a-b) = (x-b)*(4*a-0) (3*a-b)*y = 4*a*(x-b) Since the point (2/3,2/3) lies on it, we get an equation relating a and b: (3*a-b)*(2/3) = 4*a*(2/3-b) Next, since AOBC is a rhombus, the lengths of its sides must be equal. You know the coordinates of B, C, and O(0,0), and those of A are found from the parallelogram rule. Then you can compute the lengths of all the sides, and setting these expressions equal will give an additional equation relating a and b. Solve the two equations relating a and b together simultaneously, and you'll get the values of a and b. Substitute these into the expressions for the coordinates of A, and you'll be done. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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