Congruent triangle counting ...
In the grid of dots shown above, each dot has an equal vertical and horizontal spacing from the others.
A small 45-45-90 triangle has been drawn as you can see.
Counting this triangle also, how many triangles congruent to this one, of any orientation,
can be constructed from dots in this grid ?
The answer is 120.
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1 solution
Isolate 4 dots in the above grid which are separated by one unit.
Each of them can form 4 right triangles.
How many such 4 - dot squares are there ?
Ans: There are 1 + 2 + 3 + 4 + 5 + 6 + 7 such 4 dot squares.
ie. There are 28 squares. These 28 squares form ( 4 x 28 = 112) such right triangles.
Add to this number the 8 right-triangles that you find along the hypotenuse of the larger grid to give 120.