Question

How many matches would be needed to form 40 large triangles in the pattern below?

A) 120 matches
B) 180 matches
C) 200 matches
D) 240 matches

Answer 1

200 matches many matches would be needed to form 40 large triangles in the pattern below.

The correct option is C

Step-by-step explanation:

The pattern forming large triangles suggests a geometric arrangement where each large triangle requires three sides. To calculate the total number of matches needed, we can use the formula for the nth triangular number, which is given by
`\(T_n = (n \ast (n + 1))/(2)\)`. In this case, we want to find the smallest value of
`\(n\) such that \(T_n \geq 40\)`. Solving this inequality, we find
`\(n = 9\).` Now, to find the total matches needed, we multiply the number of large triangles
`(\(n\))`by 3 (since each triangle has three sides):
`\(9 \ast 3 = 27\)`.

However, this only gives us the matches forming the inner triangles. To complete the outer triangles, we need to add the matches along the perimeter, which is also the value of
`\(n\): \(27 + 9 = 36\)`. So, the total matches required to form 40 large triangles is
`\(36 \ast 3 = 108\)`. However, each match is shared by two adjacent triangles, so we need to divide this by 2 to avoid counting matches twice:
`\(108 / 2 = 54\)`.

The correct option is C