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Problem 517

AMC 12 late, AIME early

Geometry Difficulty 4.9 Find the answer

(9) If a wire of length 1 is cut into three pieces, the probability that these three pieces can exactly form a triangle is . $\qquad$

Your answer

Official solution

(9) $\frac{1}{4}$ Hint: Let the lengths of the three segments be $x, y, 1-x-y$ , then the total sample space is $x>0, y>0, x+y<1$ . The area of the region where a triangle can be formed is $x+y>\frac{1}{2}$ , area is $\frac{1}{8}$ . Therefore, the required probability is $\frac{1}{4}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.