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So I need help with this problem:

How many different non-congruent isosceles triangles can be formed by connecting three of the dots in a 4x4 square array of dots like the one shown below?

My work, is that I tried drawing different isoceles triangles and came up with the answer 9. I inputted it and it was incorrect.

Can somebody help? Thanks!

Dec 22, 2019
edited by Guest  Dec 22, 2019

#1
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This is an AoPS problem, isn't it?

I got 9 as well. But it turns out the answer is 11:

This is the solution if you want to see their reasoning:

We can construct segments of length $$1, \sqrt 2, \sqrt 5, \sqrt {10}, 2, 2\sqrt 2, \sqrt {13}, 3, 3\sqrt 2$$  (these can be obtained systematically by considering all lengths of segments from the top left dot to other dots). The following results can be obtained either by inspection or analysis of slope.

For each of $$1, \sqrt 5, \sqrt {13}, 3$$, there are zero isosceles triangles having it as its base length.
For $$\sqrt{10}$$ there is one.
For each of $$2\sqrt 2, 3\sqrt 2$$ there are two.
For each of $$2, \sqrt 2$$ there are three.
This gives a total of 11 non-congruent isosceles triangles.

Dec 22, 2019
#2
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Oh wow. Thanks dude!

Guest Dec 22, 2019