A man travels on a bicycle, 10 km east from the starting point A to reach point B, then the cycles 15 km south to reach point C. Find the shortest distance between A and C.

Options

A. 25 km
B. 5km
C. `25sqrt(13) km`
D. `5sqrt(13) km`

Explanation

To find the shortest distance between points A and C, we can visualize the problem using a right triangle. Here’s a step-by-step solution:

### Step 1: Visualize the Points
- Start by plotting the points on a coordinate system.
- Let point A be at the origin (0, 0).
- Point B will be at (10, 0) because the man travels 10 km east.
- Point C will be at (10, -15) because he then travels 15 km south from point B.

### Step 2: Determine the Coordinates
- Point A: (0, 0)
- Point B: (10, 0)
- Point C: (10, -15)

### Step 3: Calculate the Distance
To find the shortest distance between points A and C, we can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Where:
- \((x_1, y_1)\) are the coordinates of point A (0, 0)
- \((x_2, y_2)\) are the coordinates of point C (10, -15)

Substituting the coordinates into the formula:
\[ d = \sqrt{(10 - 0)^2 + (-15 - 0)^2} \]
\[ d = \sqrt{(10)^2 + (-15)^2} \]
\[ d = \sqrt{100 + 225} \]
\[ d = \sqrt{325} \]
\[ d = \sqrt{25 \times 13} \]
\[ d = 5\sqrt{13} \]

### Final Answer
The shortest distance between points A and C is \(5\sqrt{13}\) km.

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