1. How many different non-congruent isosceles triangles can be formed by connecting three of the dots in a $4\times4$ square array of dots like the one shown below? [asy] size(50); dot((0,0));dot((0,1));dot((0,2));dot((0,3)); dot((1,0));dot((1,1));dot((1,2));dot((1,3)); dot((2,0));dot((2,1));dot((2,2));dot((2,3)); dot((3,0));dot((3,1));dot((3,2));dot((3,3)); [/asy]
Two triangles are congruent if they have the same traced outline, possibly up to rotating and flipping. This is equivalent to having the same set of 3 side lengths.
2. I have $5$ different pullover shirts and $4$ different button-down shirts. In how many ways can I choose shirts for the next $9$ days if I insist on wearing pullover shirts two days in a row at least once? Assume that I wear one shirt each day, and every shirt gets worn once.
I know how to answer question 1.
1.
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 \(1 + 2 + 2 + 3 + 3 = \boxed{11}\) non-congruent isosceles triangles.
