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Part 1
The perimeter of an equilateral triangle is 15 inches more than the perimeter of a​ square, and the side of the triangle is 7 inches longer than the side of the square. Find the side of the triangle.​ (Hint: An equilateral triangle has three sides the same​ length.)   The side of the triangle is

Let's solve the problem step by step.

### Step 1: Define the variables
Let:
- \( s \) be the side length of the square.
- \( t \) be the side length of the equilateral triangle.

### Step 2: Express the perimeters
- The perimeter of the square is \( 4s \) (since all four sides of the square are equal).
- The perimeter of the equilateral triangle is \( 3t \) (since all three sides of the equilateral triangle are equal).

### Step 3: Set up the equations based on the problem
The problem states two conditions:
1. The perimeter of the equilateral triangle is 15 inches more than the perimeter of the square:
   \[
   3t = 4s + 15
   \]
2. The side of the triangle is 7 inches longer than the side of the square:
   \[
   t = s + 7
   \]

### Step 4: Solve the system of equations
We already have the equation \( t = s + 7 \). Substitute this into the first equation:

\[
3(s + 7) = 4s + 15
\]

Expand and simplify:

\[
3s + 21 = 4s + 15
\]

Subtract \( 3s \) from both sides:

\[
21 = s + 15
\]

Subtract 15 from both sides:

\[
s = 6
\]

### Step 5: Find the side of the triangle
Since \( t = s + 7 \), substitute \( s = 6 \) into the equation:

\[
t = 6 + 7 = 13
\]

### Final Answer:
The side of the triangle is **13 inches**.

Would you like more details or have any questions? Here are some related questions for further practice:

1. What would be the side of the square if the perimeter of the triangle was 10 inches more than that of the square?
2. If the perimeter of a square was increased by 20%, how would the side length of the square change?
3. What would happen to the side of the triangle if the side of the square was doubled?
4. How would the problem change if the triangle was isosceles instead of equilateral?
5. What if the side of the triangle was 5 inches shorter than the side of the square? How would that affect the solution?

**Tip:** Remember, in geometric problems, setting up the correct equations based on given conditions is crucial to solving them accurately.

Learn how to find the side length of an equilateral triangle when its perimeter is 15 inches more than that of a square. Follow a step-by-step solution to solve the problem involving geometric equations and perimeter calculations. Suitable for Grades 7-9.

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