The perimeter of an equilateral triangle is 1616 inches more than the perimeter of a​ square, and the side of the triangle is 77 inches longer than the side of the square. Find the side of the triangle.​ (Hint: An equilateral triangle has three sides the same​ length.)

Let the side length of the square be denoted as \( s \), and the side length of the equilateral triangle be denoted as \( t \).

### Step 1: Perimeter of a Square and an Equilateral Triangle
- The perimeter of a square is \( 4s \) (since it has 4 equal sides).
- The perimeter of an equilateral triangle is \( 3t \) (since all sides are the same).

According to the problem, the perimeter of the equilateral triangle is 1616 inches more than the perimeter of the square. Therefore, we can write the equation:

\[
3t = 4s + 1616
\]

### Step 2: Relationship Between the Side Lengths
We are also given that the side length of the triangle is 77 inches longer than the side length of the square. This gives us the second equation:

\[
t = s + 77
\]

### Step 3: Substitution
Substitute \( t = s + 77 \) into the first equation:

\[
3(s + 77) = 4s + 1616
\]

### Step 4: Simplification
Expand the left-hand side:

\[
3s + 231 = 4s + 1616
\]

### Step 5: Solve for \( s \)
Now, subtract \( 3s \) from both sides:

\[
231 = s + 1616
\]

Subtract 1616 from both sides:

\[
s = 231 - 1616 = -1385
\]

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The perimeter of an equilateral triangle is 1616 inches more than the perimeter of a square, and the side of the triangle is 77 inches longer than the side of the square. Find the side of the triangle. (Hint: An equilateral triangle has three sides the same length.)

Learn how to solve a perimeter problem where the perimeter of an equilateral triangle is 1616 inches more than that of a square, and the side of the triangle is 77 inches longer than the side of the square. Step-by-step solution suitable for students in Grades 8-10.

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