Consider the line containing the points and . What is the length of the hypotenuse of the right triangle formed by the intersections of and the -and -axes?
Source: NCTM Mathematics Teacher, September 2006
Points , and are the vertices of the right triangle . The length of hypotenuse equals
The slope of equals
Hypotenuse intersects the -axis at point . Using and we express the slope of as
Thus, intersects the -axis at . Observe that the four vertical lines , and intersect the -axis at equal intervals. By the Proportionality theorem, if parallel lines intersect two transversals, then they divide the transversals proportionally. Given that the four vertical lines are parallel and divide the transversal -axis into thirds, they also divide the transversal into thirds.
Using points and we express the slope of as
The triangle with vertices at and is a right triangle.