In right triangle PQR, angle R is the right angle. Point M on side PQ is connected by a line segment with point N on side RQ such that line segment MN is perpendicular to side RQ and RN = 3NQ. If MN = 48, what is the length of side PR?



Let's solve the problem step by step.

### Given Information:
1. \( \triangle PQR \) is a right triangle with \( \angle R = 90^\circ \).
2. Point \( M \) is on side \( PQ \), and point \( N \) is on side \( RQ \), such that \( MN \) is perpendicular to \( RQ \).
3. \( RN = 3NQ \) (This implies \( \frac{RN}{NQ} = 3 \)).
4. \( MN = 48 \).

### Required:
The length of side \( PR \).

### Step 1: Assign Variables
- Let \( RQ = x \).
- Then \( NQ = \frac{x}{4} \) and \( RN = \frac{3x}{4} \) since \( RN = 3 \times NQ \).
- Let \( PR = a \) and \( QR = b \).

### Step 2: Use Similar Triangles
Since \( MN \) is perpendicular to \( RQ \), \( \triangle MNR \) and \( \triangle PQR \) are similar by AA similarity criterion (since \( \angle R = \angle MNR = 90^\circ \)).

Thus, the sides of the triangles are proportional:
\[
\frac{MN}{PR} = \frac{RN}{RQ}
\]

Substituting the known values:
\[
\frac{48}{a} = \frac{\frac{3x}{4}}{x} = \frac{3}{4}
\]

### Step 3: Solve for \( PR = a \)
\[
\frac{48}{a} = \frac{3}{4}
\]
Cross-multiply:
\[
48 \times 4 = 3a
\]
\[
192 = 3a
\]
\[
a = \frac{192}{3} = 64
\]

### Answer:
The length of side \( PR \) is \( \boxed{64} \).

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Would you like more details on any of these steps? 

Here are some related questions you might consider:
1. How would the solution change if \( RN = 2NQ \) instead?
2. How does the similarity of triangles help in solving such problems?
3. What is the significance of the ratio \( \frac{RN}{NQ} = 3 \)?
4. How would you find the length of side \( PQ \)?
5. Can the Pythagorean theorem be applied to find side \( QR \)?

**Tip:** In right triangles, identifying similar triangles can simplify the calculation of unknown lengths significantly.

Learn how to solve a right triangle problem involving perpendicular segments MN and RN = 3NQ. Understand the application of similarity of triangles and proportions to find the length of side PR. Suitable for high school geometry students.

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