Points (5,0), (0,0) and (0,10) form a triangle
Point (x,y) is a point within the triangle.
What is the probability that y<x?
If (x,y) is within triangle BCO, then y<x.
Let CD = n.
Triangle BCD is similar to triangle ACO.
Thus, the legs in each triangle must be in the same ratio.
Since CO/OA = 5:10 = 1:2, CD:BD = n : 2n.
Thus, BD = 2n.
The line y=x forms a 45 degree angle with the x-axis.
Thus, in triangle BDO, BD = DO = 2n.
Thus, CO = n+2n = 3n.
Since the length of OA (10) is twice the length of CO (5), OA = 2(3n) = 6n.
Thus, BD/OA = 2n/6n = 1/3.
Since triangles BCO and ACO have the same base (CO), and the height of triangle BCO (BD=2n) is 1/3 the height of triangle ACO (OA=6n), the area of triangle BCO is 1/3 the area of triangle ACO.
Thus, P(y<x) = (area of BCO)/(area of ACO) = 1/3.
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