How many isosceles triangles can be drawn on the square grid?

Question

A point of the square grid is chosen to form an isosceles triangle together with the red segment.

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Answer 1

the answer is 5 but here is how you figure that out.



an isosceles triangle is any triangle with at least 2 equal sides so starting in the upper right corner draw 2 lines from that point back to either side of the red segment, you should be able to tell just by looking that no 2 sides are equal here. next move to the left to the next point and again draw 2 lines from this point to either end of the red segment and you should be able to notice right away that the red segment and the longer of the 2 lines you drew are equal in length. repeat again with the next dot to the left and you will notice that again one of the lines you drew is equal to the red segment. if you work your way around you will count a total of 5 isosceles triangles can be made.



beware though if the person putting the question to you is tricky they may say it is only 4, since one of the 5 is an "isosceles right triangle".

Answer 2

Isosceles Square

Answer 3

To count triangles join all diagonals.

number to each trianle from 1 to 64 as here are 64 small triangles.



consider 1 × 1 square there are (4 + 4) triangles and total squares are 16

total triangles = 16 × 8 = 128



consider 2 × 2 square there are 9 such squares

consider triangles with 4 numbers = there are 4 triangles in 2 × 2 square.

total triangles = 9 × 4 = 36



consider 3 × 3 square there are 4 such squares

consider triangles with 8 numbers = there are 4 triangles in 3 × 3 square.

total triangles = 4 × 4 = 16

consider triangles with 18 numbers = there are 2 triangles in 3 × 3 square.

total triangles = 4 × 2 = 8



consider 4 × 4 square there is1 such squares

consider triangles with 32 numbers = there are 2 triangles in 4 × 4 square.

total triangles = 1 × 2 = 2



total triangles = 128 + 36 + 16 + 8 + 2 = 190

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Answer 4

eight.