Question

Math

Posted 4 months ago

`Consider a segment of length 10. Points A and B are chosen randomly such that A and B divide the segment into three smaller segments. What is the probability that the three smaller segments could form the sides of a triangle?`

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

Consider a segment of length 10. To form a triangle, the sum of the lengths of any two segments must be greater than the length of the third segment. This is known as the triangle inequality

step 2

Let the first point A divide the segment into two parts of lengths x and 10 - x. The second point B will divide one of these parts further. Without loss of generality, assume B divides the segment of length x into parts of lengths y and x - y

step 3

The three segments formed are of lengths y, x - y, and 10 - x. The triangle inequalities that must be satisfied are: y + (x - y) > 10 - x, y + (10 - x) > x - y, and (x - y) + (10 - x) > y

step 4

Simplifying the inequalities, we get: x > 5, 10 - x > 5, and x - 2y > 0

step 5

The inequalities imply that x must be between 5 and 10, and y must be between 0 and x/2

step 6

The geometric probability is the area of the region satisfying the triangle inequalities divided by the total area of the possible region where points A and B can be chosen

step 7

The total possible area is a triangle with base 10 and height 5, which corresponds to the case where 0 < x < 10 and 0 < y < x

step 8

The area satisfying the triangle inequalities is a triangle with base 5 and height 2.5

step 9

Calculate the area of the triangle representing all possible outcomes: A_{total} = \frac{1}{2} \times 10 \times 5 = 25

step 10

Calculate the area of the triangle representing the successful outcomes: A_{success} = \frac{1}{2} \times 5 \times 2.5 = 6.25

step 11

The probability is the ratio of the successful area to the total area: P = \frac{A_{success}}{A_{total}} = \frac{6.25}{25}

step 12

Simplify the probability: P = \frac{6.25}{25} = \frac{1}{4}

Answer

The probability that the three smaller segments could form the sides of a triangle is $\frac{1}{4}$.

Key Concept

Geometric Probability and Triangle Inequality

Explanation

The problem is solved using geometric probability, considering the area of successful outcomes over the total possible area, while ensuring the triangle inequalities are satisfied.

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