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).