The problem consists of two parts. Part 1: There are three different equilateral triangles made of circles. The number of circles is calculated in three ways using the formulas $(6-1) \times 3$, $6 \times 3 - 3$, and $(6-2) \times 3 + 3$. We need to match each formula to the corresponding triangle, A, B, or C, by figuring out which visual representation the formula corresponds to. Part 2: We need to express the formulas above using the variable $a$, where $a$ represents the number of circles on one side of the equilateral triangle.

1. Problem Description

The problem consists of two parts.

Part 1: There are three different equilateral triangles made of circles. The number of circles is calculated in three ways using the formulas $(6-1) \times 3$, $6 \times 3 - 3$, and $(6-2) \times 3 + 3$. We need to match each formula to the corresponding triangle, A, B, or C, by figuring out which visual representation the formula corresponds to.

Part 2: We need to express the formulas above using the variable $a$, where $a$ represents the number of circles on one side of the equilateral triangle.

2. Solution Steps

Part 1:

Let's evaluate each expression:

$(6-1) \times 3 = 5 \times 3 = 15$. This corresponds to triangle B. Each side is made of 6 circles but we count the corner circles twice which results in this calculation, so $6 \times 3 = 18$, and subtract

3.

$6 \times 3 - 3 = 18 - 3 = 15$. This corresponds to triangle A. Each side has 6 circles, and we multiply by 3 to get 18, but we counted each corner twice, so subtract

3.

$(6-2) \times 3 + 3 = 4 \times 3 + 3 = 12 + 3 = 15$. This corresponds to triangle C. It means that on each side we exclude 2 and multiply the rest by

3. At the end, we add 3 to compensate the excluded one on each corner of the triangle.

Part 2:

We need to rewrite the expressions using the variable $a$. In all three cases, the number of circles per side is $a$.

The number of circles is

1

5. Therefore, $a=6$.

Expression 1: $(6-1) \times 3$

Replacing 6 with $a$, we get $(a-1) \times 3$

Expression 2: $6 \times 3 - 3$

Replacing 6 with $a$, we get $a \times 3 - 3 = 3a-3$

Expression 3: $(6-2) \times 3 + 3$

Replacing 6 with $a$, we get $(a-2) \times 3 + 3$

3. Final Answer

Part 1:

$(6-1) \times 3$: B

$6 \times 3 - 3$: A

$(6-2) \times 3 + 3$: C

Part 2:

$(a-1) \times 3$

$3a-3$

$(a-2) \times 3 + 3$