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We are given a quadrilateral \( PQRS \), where:

- \( PS = 12 \)
- \( PQ = 4y - 1 \)
- \( RS = 3y \)
- \( QR = y + 3 \)
- The perimeter is 78 units.

We are also given that \( QR = 11 \), so we can set \( y + 3 = 11 \), and solve for \( y \).

### Step 1: Solve for \( y \)
\[
y + 3 = 11
\]
\[
y = 11 - 3 = 8
\]

### Step 2: Substitute \( y = 8 \) into the expressions for the other sides
- \( PQ = 4y - 1 = 4(8) - 1 = 32 - 1 = 31 \)
- \( RS = 3y = 3(8) = 24 \)

### Step 3: Verify the perimeter
The perimeter is the sum of all sides:
\[
PS + PQ + RS + QR = 12 + 31 + 24 + 11 = 78 \text{ units}
\]
This confirms that the calculation is correct, as the total is 78 units.

Thus, the length of side \( QR \) is **11 units** (as given), and the value of \( y \) is **8**.

Would you like further details or clarifications?

Here are five related questions:
1. How is the perimeter of a quadrilateral calculated?
2. What is the significance of solving for \( y \) in this type of problem?
3. How does knowing one side's length help in solving the rest of the problem?
4. What are the steps to verify if a solution for \( y \) is correct in geometric problems?
5. How can this method be applied to solve for unknowns in different polygon types?

**Tip:** Always double-check your calculations by verifying with the given conditions (like the perimeter).

The perimeter of the quadrilateral below is 78 units. Find the length of side QR.

Solve the perimeter of a quadrilateral problem, where the total perimeter is 78 units, and one side's length is unknown. Learn how to use algebraic equations to solve for side QR and verify your answer. Suitable for students in Grades 7-9.

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