Let's analyze and solve the problem:

We are tasked with finding the length of $PR$ in $\triangle PQR$, where the given data is as follows:

• $PQ = 39 \, \text{cm}$,
• $\overline{PN}$ is an altitude, meaning $PN \perp QR$,
• $QN = 36 \, \text{cm}$,
• $RN = 8 \, \text{cm}$.

Step 1: Understand the problem

Since $PN$ is the altitude of the triangle, $QN$ and $RN$ are segments of the base $QR$: $QR = QN + RN = 36 \, \text{cm} + 8 \, \text{cm} = 44 \, \text{cm}.$

We are finding $PR$ using the Pythagorean theorem.

Step 2: Use the Pythagorean theorem

The Pythagorean theorem applies here because $PN$ is perpendicular to $QR$. This splits $\triangle PQR$ into two right triangles: $\triangle PQR$ and $\triangle PRN$.

1. Compute the altitude $PN$: Using the relationship of right triangles: $PQ^2 = PN^2 + QN^2$ Substitute $PQ = 39$ and $QN = 36$: $39^2 = PN^2 + 36^2$ $1521 = PN^2 + 1296$ $PN^2 = 1521 - 1296 = 225$ $PN = \sqrt{225} = 15 \, \text{cm}.$

2. Calculate $PR$: Using $\triangle PRN$ and applying the Pythagorean theorem again: $PR^2 = PN^2 + RN^2$ Substitute $PN = 15$ and $RN = 8$: $PR^2 = 15^2 + 8^2$ $PR^2 = 225 + 64$ $PR^2 = 289$ $PR = \sqrt{289} = 17 \, \text{cm}.$

Final Answer:

$PR = 17 \, \text{cm}.$

Would you like a detailed explanation of any step or have more questions?

Here are 5 related questions for practice:

1. How is the Pythagorean theorem used in solving right triangles?
2. If $QR = 50$ and $PQ = 40$, can you find the altitude $PN$?
3. Why is $PN$ perpendicular to $QR$ in this problem?
4. What is the significance of the altitude in triangle geometry?
5. How would this problem change if $QN$ and $RN$ were unknown?

Tip: Always label the triangle and use diagrams to visualize relationships between sides and angles.