Splitting Triangles
Point A has the coordinates (4,37). This point forms a corner of a triangle with B and C, both of which lie on the line 3y-2x=12 and are equidistant from A, forming an isosceles triangle. If px+qy=r is the line of reflection of this triangle, for coprime positive integers p,q,r, what is p+q+r?
The answer is 91.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
The line which cuts the isosceles in half will pass through a and be perpendicular to 3y-2x=12 by nature. N.B. it does not matter where points B and C specifically are on the line. Thus the gradient of the perpendicular will be the negative reciprocal of the other line, which has a gradient of 2/3. So the gradient of the perpendicular will be -1.5. Then, using either y=mx+c (37=-1.5(4)+c) or y-37=-1.5(x-4), we can rearrange this to get 3x+2y=86. And so 3+2+86=91.