How many triangles #7

1.) How many triangles are there in this hexagon?

how many triangles in this hexagon

Answer with solution

Again it is good to use symmetry here, we can brake this image into six small triangles each formed by one of the side of the hexagon and each of the triangle is divided in half by a line. we will count the number of triangles formed by each part and by taking two or more such parts together.

Number of triangles formed by one part: 3 so total 3*6 = 18.
Number of triangles formed by taking two adjacent parts together: 2*3(three such cases as for 3 cases it is zero) so total 6.
Number of triangles formed by taking three adjacent triangles: 2*6 ( for each case 2 triangles are possible) so total 12
Number of triangles formed by taking 4 or 5 parts together: 0
Number of triangles formed by taking all 6 parts together: 1

Thus, total number of triangles possible in above puzzle are: 18+6+12+1= 37