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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?

 Apr 27, 2023
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The area of triangle XYD is at most 24 if and only if the height of triangle XYD is at most 2. The height of triangle XYD is at most 2 if and only if point D is within the shaded region below:

[asy] unitsize(1 cm);

pair X, Y, Z, D;

X = (0,0); Y = (12,0); Z = (6,6sqrt(3)); D = (6,3sqrt(3));

draw(X--Y--Z--X); draw(rightanglemark(X,Y,D,10)); draw(rightanglemark(Y,D,Z,10)); draw(X--D); draw(D--Y);

label("X", X, SW); label("Y", Y, NE); label("Z", Z, NW); label("D", D, S); [/asy]

The shaded region is a right triangle with legs of length 6 and 3, so its area is 1/2(6)(3)=9. Therefore, the probability that the area of triangle XYD is at most 24 is 9/12 = 3/4​​.

 Apr 27, 2023

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