A, B, C, D, are four towns at the vertices of a rectangle. W is a well inside the rectangle. If the shortest distance to the well from the towns, A, B, C is 4 km, 5 km and 6 km respectively, find the shortest distance to the well from town D.
(a)7 km
(b)`4sqrt(2) km`
(c)9 km
(d)`3sqrt(3) km`

Options

A. 7 km
B. `4sqrt(2) km`
C. 9 km
D. `3sqrt(3) km`

Answer

To solve the problem, we need to find the shortest distance from town D to the well W inside the rectangle ABCD, given the distances from towns A, B, and C to W.

### Step-by-Step Solution:

1. **Identify the Given Distances**:
- Distance from A to W (AW) = 4 km
- Distance from B to W (BW) = 5 km
- Distance from C to W (CW) = 6 km
- Distance from D to W (DW) = x km (unknown)

2. **Use the Distance Formula**:
We can use the formula for distances in a rectangle:
\[ AW^2 + CW^2 = BW^2 + DW^2 \]
This formula holds true for points inside the rectangle.

3. **Substitute the Known Values**:
Plugging in the values we have:
\[ (4)^2 + (6)^2 = (5)^2 + x^2 \]

4. **Calculate the Squares**:
Calculate each term:
\[ 16 + 36 = 25 + x^2 \]
Simplifying gives:
\[ 52 = 25 + x^2 \]

5. **Isolate x^2**:
Rearranging the equation to solve for \( x^2 \):
\[ 52 - 25 = x^2 \]
\[ 27 = x^2 \]

6. **Solve for x**:
Taking the square root of both sides:
\[ x = \sqrt{27} = 3\sqrt{3} \]

7. **Conclusion**:
The shortest distance from town D to the well W is \( 3\sqrt{3} \) km.

### Final Answer:
The shortest distance from town D to the well W is \( 3\sqrt{3} \) km, which corresponds to option (d).

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