`Q` is a point in the interior of a rectangle `ABCD`. If `QA = 3 cm, QB = 4 cm and QC = 5 cm`, then the length of QD (in cm) is

Options

A. `3sqrt(2)`
B. `5sqrt(2)`
C. `sqrt(34)`
D. `sqrt(41)`

Answer

To solve the problem, we will use the relationship that holds for a point \( Q \) inside rectangle \( ABCD \). The relationship states that:

\[ QA^2 + QC^2 = QB^2 + QD^2 \]

Given:
- \( QA = 3 \, \text{cm} \)
- \( QB = 4 \, \text{cm} \)
- \( QC = 5 \, \text{cm} \)

We need to find \( QD \).

### Step-by-Step Solution:

1. **Square the lengths of \( QA \), \( QB \), and \( QC \)**:
\[ QA^2 = 3^2 = 9 \]
\[ QB^2 = 4^2 = 16 \]
\[ QC^2 = 5^2 = 25 \]

2. **Substitute the squared values into the relationship**:
\[ QA^2 + QC^2 = QB^2 + QD^2 \]
Substitute the values we calculated:
\[ 9 + 25 = 16 + QD^2 \]

3. **Combine the left side**:
\[ 34 = 16 + QD^2 \]

4. **Isolate \( QD^2 \)**:
\[ QD^2 = 34 - 16 \]
\[ QD^2 = 18 \]

5. **Take the square root to find \( QD \)**:
\[ QD = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \, \text{cm} \]

### Final Answer:
The length of \( QD \) is \( 3\sqrt{2} \, \text{cm} \).

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