SOLUTION: An equilateral triangle ABC circumscribes the circle with equation x^2 + y^2 = a^2. The side BC of the triangle has equation x=-1 a) Find the equations of AR and AC. b) Find t

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: An equilateral triangle ABC circumscribes the circle with equation x^2 + y^2 = a^2. The side BC of the triangle has equation x=-1 a) Find the equations of AR and AC. b) Find t      Log On


   

Question 1096936: An equilateral triangle ABC circumscribes the circle with equation x^2 + y^2 = a^2. The side BC of the triangle has equation x=-1
a) Find the equations of AR and AC.
b) Find the equation of the circle circumscribing triangle ABC.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!

From the given information, the center of the circle is O(0,0). Since side BC of the circumscribed equilateral triangle is on the line x=-1, the point of tangency of the circle with side BC is P(-1,0).

Let B be the vertex of the triangle with a positive y coordinate and C be the vertex with a negative y coordinate.

Triangles BPO and CPO are 30-60-90 right triangles, so BP = CP = a*sqrt(3) and OA = OB = OC = 2a. So B is (-1,a*sqrt(3)) and C is (-1,-a*sqrt(3)).

And since the medians of a triangle divide each other in the ratio 1:2, we know A is (2,0).

Since we now have the coordinates of all three vertices of the triangle, and since we know OB is the radius of the circle circumscribed about triangle ABC, we can answer all parts of the problem.