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Let's break down the problem step by step:

### Part (a): 
You are asked to determine the coordinates of point **R** and the value of **a**.
From the diagram, we can see the parallelogram PQRS is drawn on the Cartesian plane with given points:
- \( P(3, -1) \)
- \( Q(7, 1) \)
- \( S(a, -5) \)

The property of a parallelogram states that the diagonals bisect each other. Hence, the midpoint of diagonal **PR** must be equal to the midpoint of diagonal **QS**.

#### Step 1: 
Find the midpoint of diagonal **PR**.

- The midpoint of \( P(3, -1) \) and \( R(x_2, y_2) \) is:
  \[
  \left( \frac{3 + x_2}{2}, \frac{-1 + y_2}{2} \right)
  \]

#### Step 2: 
Find the midpoint of diagonal **QS**.

- The midpoint of \( Q(7, 1) \) and \( S(a, -5) \) is:
  \[
  \left( \frac{7 + a}{2}, \frac{1 - 5}{2} \right) = \left( \frac{7 + a}{2}, -2 \right)
  \]

#### Step 3: 
Set the midpoints equal to each other.

Since the midpoints of diagonals PR and QS must be equal, we have:
\[
\frac{3 + x_2}{2} = \frac{7 + a}{2} \quad \text{and} \quad \frac{-1 + y_2}{2} = -2
\]

From the second equation:
\[
\frac{-1 + y_2}{2} = -2 \implies -1 + y_2 = -4 \implies y_2 = -3
\]

From the first equation:
\[
\frac{3 + x_2}{2} = \frac{7 + a}{2} \implies 3 + x_2 = 7 + a \implies x_2 = 4 + a
\]

Thus, point **R** has coordinates \( (4 + a, -3) \).

---

### Part (b): 
Without finding the coordinates of point **Q**, you are asked to calculate the length of diagonal **QS**.

We can use the distance formula for the points \( Q(7, 1) \) and \( S(a, -5) \):

\[
d = \sqrt{(a - 7)^2 + (-5 - 1)^2} = \sqrt{(a - 7)^2 + (-6)^2}
\]
\[
d = \sqrt{(a - 7)^2 + 36}
\]

This gives the length of diagonal **QS** as \( \sqrt{(a - 7)^2 + 36} \).

---

Would you like further details on any specific part or have any questions?

Here are five related questions to expand on the above:

1. How do you determine the midpoint of a line segment on a Cartesian plane?
2. What are the properties of a parallelogram that are useful in coordinate geometry?
3. How can we solve for the unknown value of \( a \) using the given conditions?
4. What is the distance formula, and how is it derived?
5. How can you verify the coordinates of point R once \( a \) is known?

**Tip:** When dealing with geometry on the Cartesian plane, always check if the shapes involve properties like midpoints, perpendicularity, or parallelism to simplify calculations.

Determine the coordinates of point R and the value of a. Without finding the coordinates of point Q, calculate the length of diagonal QS.

Learn how to determine the coordinates of a point R and the value of a, as well as calculate the length of diagonal QS without finding Q's coordinates, using key properties of parallelograms and distance formulas. Suitable for students in Grades 9-11.

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