How Many Right Triangles Can You Find?

m

2

n

2

partial hint 1

partial hint 2

solution

This Demonstration shows the number of right triangles that have their vertices at the points of an

m×n

regular grid (so that the horizontal and vertical distance between points is the same). With "solution" checked, the number of triangles that have a right angle at the point

(i,j)

,

i=1,2,…,m

,

j=1,2,…,n

is shown.

For example, for a 2x3 regular grid, the vertices of the 14 right triangles are:

{1,1},{1,2},{2,1}{1,1},{1,3},{2,1}

{1,1},{1,2},{2,2}{1,3},{1,2},{2,2}

{2,1},{1,2},{2,3}

{1,2},{1,3},{2,3}{1,1},{1,3},{2,3}

{1,1},{2,1},{2,2}{1,1},{2,1},{2,3}

{2,1},{2,2},{1,2}{1,2},{2,2},{2,3}{1,1},{2,2},{1,3}

{1,3},{2,3},{2,2}

{1,3},{2,3},{2,1}