Track / Stage 4 / 117 of 340 #377 of 2000

Problem 377

AMC 12 late, AIME early

Geometry Difficulty 4.7 Find the answer

4. One side of a triangle is 2, the median of this side is 1, and the sum of the other two sides is $1+\sqrt{3}$ , then the area of this triangle is:
(A) 1;
(B) $\frac{1}{2} \sqrt{3}$ ;
(C) $\sqrt{3}$ ;
(D)

Cannot be determined.

Your answer

Official solution

Answer: (B).
From the given side of the triangle being 2, and the median to this side being 1, we can determine that this triangle is a right triangle, with the hypotenuse being 2. Since the sum of the two legs is $1+\sqrt{3}$ , we can find that one leg is 1 and the other is $\sqrt{3}$ . Therefore, the area of this triangle is $\frac{1}{2} \sqrt{3}$ .

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