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This question involves two chords, \(PQ\) and \(QR\), on a circle of radius 7 units, with each chord having a length of 7 units. The problem asks for the length of the chord \(PR\), which spans between points \(P\) and \(R\).

To solve this, let's break it down:

### Step 1: Visualizing the geometry
The chords \(PQ\) and \(QR\) are each 7 units in length. The center of the circle is \(O\), and the points \(P\), \(Q\), and \(R\) lie on the circumference of the circle. Since \(PQ = QR = 7\), we can think of this as an isosceles triangle with \(OQ\) as one of the radii of the circle, and thus \(OQ = 7\) as well.

This triangle is symmetric because both chords are of equal length, meaning that the angle \(\angle POR\) between the two radii \(OP\) and \(OR\) should be the same.

### Step 2: Use the cosine rule
To find the length of the chord \(PR\), we can use the cosine rule in triangle \(OPR\), where \(O\) is the center of the circle:

\[
PR^2 = OP^2 + OR^2 - 2 \cdot OP \cdot OR \cdot \cos(\theta)
\]
Where:
- \(PR\) is the chord we want to find,
- \(OP = OR = 7\) (since these are radii),
- \(\theta\) is the angle between \(OP\) and \(OR\).

### Step 3: Find the angle between the radii
Since \(PQ = QR = 7\), the points \(P\), \(Q\), and \(R\) divide the circle into equal parts, so \(\theta = 120^\circ\). This is because the angle subtended by equal chords in a circle at the center will be equal. The total angle around the center of the circle is \(360^\circ\), and the chords divide it into thirds: \(360^\circ / 3 = 120^\circ\).

### Step 4: Apply cosine rule
Substituting into the cosine rule:
\[
PR^2 = 7^2 + 7^2 - 2 \cdot 7 \cdot 7 \cdot \cos(120^\circ)
\]

We know \(\cos(120^\circ) = -\frac{1}{2}\), so:
\[
PR^2 = 49 + 49 - 2 \cdot 7 \cdot 7 \cdot \left(-\frac{1}{2}\right)
\]
\[
PR^2 = 98 + 49 = 147
\]

Therefore, the length of \(PR\) is:
\[
PR = \sqrt{147} = \sqrt{49 \times 3} = 7\sqrt{3}
\]

### Step 5: Simplify
To match the answer choices, we need to simplify \(7\sqrt{3}\):
\[
PR = \frac{21}{\sqrt{3}}
\]

Thus, the length of the chord \(PR\) is \(\frac{21}{\sqrt{3}}\), which corresponds to option **D**.

### Final Answer:
The correct answer is \(\boxed{D}\).

---

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

1. How does the cosine rule apply to triangles in a circle?
2. What is the relationship between equal chords and the angles they subtend at the center of a circle?
3. How can you derive the formula for the length of a chord given the radius and the central angle?
4. How would the solution change if the length of the chords were different?
5. What other methods could be used to find the length of chord \(PR\) geometrically?

**Tip:** When working with circles, always consider how symmetry and the properties of central angles can simplify complex-looking problems.

On a circle of radius 7 units, PQ and QR are chords of length 7 units each. What is the length of the chord PR in units?

This problem explores the geometric properties of a circle with radius 7 units, where two chords PQ and QR are equal in length. Learn how to apply the cosine rule to calculate the length of the chord PR, which spans between points P and R on the circle.

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