In `Delta PQR, PQ = x, QR = x + 3`, and `PR = y`. If `x = y + 3`, then which one of the following is true?

Options

A. `/_P lt /_Q lt /_R`
B. `/_Q lt /_R lt /_P`
C. `/_R lt /_P lt /_Q`
D. `/_P lt /_R lt /_Q`

Answer

To solve the problem step by step, we will analyze the given information about triangle PQR and derive the relationships between the sides and angles.

### Step 1: Identify the sides of the triangle
Given:
- \( PQ = x \)
- \( QR = x + 3 \)
- \( PR = y \)

### Step 2: Substitute the value of \( x \)
We are given that \( x = y + 3 \). We will substitute this value into the expressions for the sides of the triangle.

1. For \( PQ \):
\[ PQ = x = y + 3 \]

2. For \( QR \):
\[ QR = x + 3 = (y + 3) + 3 = y + 6 \]

3. For \( PR \):
\[ PR = y \]

Now we have the sides expressed in terms of \( y \):
- \( PQ = y + 3 \)
- \( QR = y + 6 \)
- \( PR = y \)

### Step 3: Determine the order of the sides
To find out which side is the longest and which is the shortest, we compare the expressions:
- \( PR = y \)
- \( PQ = y + 3 \)
- \( QR = y + 6 \)

Since \( y \) is positive (as sides of a triangle must be positive), we can conclude:
- \( PR < PQ < QR \)
- Therefore, the order of the sides is:
- Smallest: \( PR \)
- Middle: \( PQ \)
- Largest: \( QR \)

### Step 4: Relate the sides to the angles
According to the triangle inequality, the angle opposite the longest side is the largest angle, and the angle opposite the shortest side is the smallest angle. Thus:
- Angle opposite \( PR \) (smallest side) is \( \angle Q \)
- Angle opposite \( PQ \) (middle side) is \( \angle R \)
- Angle opposite \( QR \) (largest side) is \( \angle P \)

### Step 5: Order the angles
From the above relationships, we can conclude:
- \( \angle Q < \angle R < \angle P \)

### Conclusion
Thus, the correct answer is:
\[ \angle Q < \angle R < \angle P \]