본고사 확률 문제 풀이_by chatGPT

Q. There is a square ABCD. Find the probability that triangle PBC is an obtuse triangle by taking any point P inside this square.

 

A.

To find the probability that triangle PBC is obtuse, we need to consider all possible triangles that can be formed by taking a point P inside the square and determining the number of those triangles that are obtuse.

There are two ways that triangle PBC can be obtuse: either the angle at B is obtuse, or the angle at C is obtuse.

If the angle at B is obtuse, then point P must be located in the region below the line segment BC. This region is a parallelogram, and the probability that point P falls within this region is equal to the area of the parallelogram divided by the area of the square. The area of the parallelogram is equal to the product of the base (BC) and the height (BD), which is aa. The area of the square is aa, so the probability that point P falls within this parallelogram is aa/(aa) = 1.

If the angle at C is obtuse, then point P must be located in the region to the left of the line segment PC. This region is also a parallelogram, and the probability that point P falls within this region is equal to the area of the parallelogram divided by the area of the square. The area of the parallelogram is equal to the product of the base (PC) and the height (CD), which is aa. The area of the square is aa, so the probability that point P falls within this parallelogram is aa/(aa) = 1.

Since the two cases are exclusive (i.e., point P cannot be in both regions at the same time), we can add the probabilities to find the total probability that triangle PBC is obtuse. The probability that triangle PBC is obtuse is 1 + 1 = 2.

However, this probability is greater than 1, which is not possible. This means that our calculation is incorrect, and we need to reconsider our approach.

One possibility is that we have double-counted some triangles. For example, if point P is located at the intersection of the two parallelograms (i.e., the center of the square), then triangle PBC is both an obtuse triangle with the angle at B being obtuse and an obtuse triangle with the angle at C being obtuse. In this case, we have counted the same triangle twice in our calculation.

To correct for this double-counting, we need to subtract the number of triangles that are counted twice from our total. The only way that a triangle can be counted twice is if point P is located at the intersection of the two parallelograms. This region is a square with side length a/2, and the probability that point P falls within this region is equal to the area of the square divided by the area of the larger square. The probability that point P falls within this region is (a/2)(a/2)/(aa) = 1/4.

Subtracting the probability of double-counting from our total probability gives us a final probability of 2 - 1/4 = 7/4. This probability is still greater than 1, which is not possible.

Upon further examination, we can see that our initial assumption that the probability of an obtuse triangle is equal to the area of the region where point P must be located is incorrect. This is because the area of the region where point P must be located depends on the length of the sides of the triangle, and the length of the sides of the triangle

 

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