If two points are selected at random from the interval , what is the probability that the distance between them is less than one fourth?

Source: NCTM Mathematics Teacher, February 2006

**SOLUTION**

**
**Consider two points and on the interval. Create an isosceles right triangle by drawing a perpendicular of length to .

Keeping the length of the segment at a fixed value of , move the segment from left to right until is at and at . As the segment moves from left to right the isosceles right triangle moves within the confines of the trapezoid from the lower left corner to the upper right hand corner ending at the location of triangle .

What we have done is convert the problem of two points and on the interval to the problem of an isosceles right triangle moving inside the trapezoid . The probability that the distance between and is less than one fourth equals the probability that the isosceles right triangle stays within the confines of the trapezoid .

The sample space is the isosceles right triangle of side length . Its area equals .

The figure shows that the trapezoid is made up of isosceles triangles .

Area of trapezoid =

Probability that the distance between and is less than equals

**Answer**:

*Alternative solution*

Let and two points on the interval . We want .

The diagonal from to in the above figure is the set of points and such that . The line above the diagonal is the set of points and the line under it is the set of points . As long as the coordinates stay inside the region delineated by these two lines, .

The square of side represents the sample space; its area equals .

Area of the region = area of square – area of two isosceles right triangles of side length

Probability that the distance between and is less than equals