The construction of a `DeltaPQR` in which PQ = 7 cm, `angleP = 45^@` is possible when (QR + PR) is

Options

A. 6 cm
B. 7 cm
C. 8 cm
D. 5 cm

Answer

To determine the condition under which the triangle \( \Delta PQR \) can be constructed with the given parameters, we follow these steps:

### Step 1: Understand the Triangle Inequality Theorem
The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

### Step 2: Identify the Given Values
In this problem, we are given:
- \( PQ = 7 \) cm
- \( \angle P = 45^\circ \)

We need to find the condition for \( QR + PR \).

### Step 3: Apply the Triangle Inequality
According to the Triangle Inequality Theorem, we have:
1. \( PQ + QR > PR \)
2. \( PQ + PR > QR \)
3. \( QR + PR > PQ \)

Since we know \( PQ = 7 \) cm, we can rewrite the inequalities:
1. \( 7 + QR > PR \)
2. \( 7 + PR > QR \)
3. \( QR + PR > 7 \)

### Step 4: Focus on the Relevant Inequality
The most relevant inequality for our question is:
\[ QR + PR > 7 \]

### Step 5: Determine the Minimum Value for \( QR + PR \)
From the inequality \( QR + PR > 7 \), we can conclude that the sum \( QR + PR \) must be greater than 7 cm.

### Step 6: Identify Possible Values
If we consider integer values greater than 7, the smallest integer that satisfies this condition is 8. Therefore, \( QR + PR \) must be at least 8 cm for the triangle to be constructed.

### Conclusion
Thus, the construction of triangle \( \Delta PQR \) is possible when \( QR + PR \) is greater than 7 cm, and the minimum value for \( QR + PR \) that satisfies this condition is 8 cm.

### Final Answer
The correct option is 8 cm.

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