Right triangle XYZ has legs of length XY = 12 and YZ = 6. Point D is chosen at random within the triangle XYZ. What is the probability that the area of triangle XYD is at most 24?
The area of triangle XYD is at most 24 if and only if the height of triangle XYD is at most 4. The height of triangle XYD is at most 4 if and only if point D is within the shaded region below.
[asy] unitsize(0.5 cm);
pair X, Y, Z, D;
X = (0,0); Y = (12,0); Z = (6,6*sqrt(3)); D = (6,3);
draw(X--Y--Z--cycle); draw(X--D);
label("X", X, SW); label("Y", Y, SE); label("Z", Z, NW); label("D", D, S);
draw((0,3)--(12,3)); [/asy]
The shaded region is a right triangle with legs of length 6 and 3. The area of this triangle is (6)(3)/2=9. Therefore, the probability that the area of triangle XYD is at most 24 is the ratio of the area of the shaded region to the area of triangle XYZ, which is 9/(12)(6)=1/8.
