Question

Fuzzy draws a segment of positive length in a plane. How many locations can Fuzzy place another point in the same plane to form a non-degenerate isosceles right triangle with vertices consisting of his new point and the endpoints of the segment?

Answer 1

Answer: 2 locations.

Step-by-step explanation:

This segment will be the hypotenuse, now, if we find the exact middle of this segment and we draw a line that cuts perpendicularly the segment by the middle, then we can put a point in any point of that line (except in the segment because this will make a degenerate triangle). Then we connect both extremes of the segment with that point, and for how we find it, we know that these new lines will have the same lenght, so this will be an isosceles triangle.

Now, if we want that the triangle is also a right triangle, then the angle between the new sides must be 90°, if we put the point near the segment, the angle will be larger than 90°, and if we put it really far away, the angle will be smaller than 90°. So for each side of this line, we have only one point where the angle is exactly 90°.

this means that we have 2 locations that can create a non-degenerate isosceles right triangle.