To determine the perimeter of the given figure, we analyze the triangle $PQR$, where:

• $QD = 6 \, \text{cm}$
• $DR = 3 \, \text{cm}$
• $PD = 4 \, \text{cm}$ (height of $P$ to base $QR$)
• $QR = QD + DR = 6 \, \text{cm} + 3 \, \text{cm} = 9 \, \text{cm}$

We now calculate the lengths $PQ$ and $PR$ using the Pythagorean Theorem.

Step 1: Find $PQ$

The right triangle $PQD$ has:

• $PD = 4 \, \text{cm}$ (height),
• $QD = 6 \, \text{cm}$ (base segment).

$PQ = \sqrt{(QD)^2 + (PD)^2} = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} \approx 7.21 \, \text{cm}$

Step 2: Find $PR$

The right triangle $PDR$ has:

• $PD = 4 \, \text{cm}$ (height),
• $DR = 3 \, \text{cm}$ (base segment).

$PR = \sqrt{(DR)^2 + (PD)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{cm}$

Step 3: Calculate the Perimeter

The perimeter of $\triangle PQR$ is:

$\text{Perimeter} = PQ + PR + QR$

Substitute the known values: $\text{Perimeter} = 7.21 \, \text{cm} + 5 \, \text{cm} + 9 \, \text{cm} = 21.21 \, \text{cm}$

Final Answer:

The perimeter of the triangle is 21.21 cm.
Correct option: B.

Would you like a further breakdown of these steps? Let me know! Here are some related questions to explore:

1. How do you apply the Pythagorean Theorem to find unknown sides of a triangle?
2. What is the formula for the perimeter of a triangle?
3. How can the distance between two points in a plane be calculated?
4. How would the perimeter change if $PD$ was longer?
5. Can you generalize the Pythagorean Theorem for larger triangles?

Tip: Always check for right triangles when given perpendicular heights or segments. It simplifies calculations!