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?

Guest Jun 3, 2023

#1**0 **

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.

Guest Jun 3, 2023