In a triangle PQR, PQ=20 cm and PR =6 cm, the side QR is :

Options

A. equal to 14 cm
B. less than 14 cm
C. greater than 14 cm
D. none of the above

Answer

To determine the possible lengths of side QR in triangle PQR, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given:
- PQ = 20 cm
- PR = 6 cm
- Let QR = x (the length we want to find)

According to the triangle inequality theorem, we have the following inequalities:

1. PQ + PR > QR
2. PQ + QR > PR
3. PR + QR > PQ

Now, we will apply these inequalities one by one:

### Step 1: Apply the first inequality
\[ PQ + PR > QR \]
\[ 20 + 6 > x \]
\[ 26 > x \]
This means:
\[ x < 26 \]

### Step 2: Apply the second inequality
\[ PQ + QR > PR \]
\[ 20 + x > 6 \]
\[ x > 6 - 20 \]
\[ x > -14 \]
Since length cannot be negative, we can ignore this inequality for practical purposes.

### Step 3: Apply the third inequality
\[ PR + QR > PQ \]
\[ 6 + x > 20 \]
\[ x > 20 - 6 \]
\[ x > 14 \]

### Conclusion
From the inequalities derived:
1. \( x < 26 \)
2. \( x > 14 \)

Thus, the length of side QR must be greater than 14 cm and less than 26 cm.

### Final Answer
The side QR is greater than 14 cm and less than 26 cm.

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