Point A is 10 miles West of point B. point B is 30 miles North of Point C. Point C is 20 miles East of Point D. What is the distance between point A and D ?

Options

A. `10sqrt(10)` miles
B. `10sqrt(20)` mile
C. `20sqrt(10)` miles
D. `30sqrt(10)` miles

Answer

To find the distance between point A and point D, we can follow these steps:

### Step 1: Understand the positions of the points
- Point A is 10 miles west of point B.
- Point B is 30 miles north of point C.
- Point C is 20 miles east of point D.

### Step 2: Set up a coordinate system
Assume:
- Point D is at the origin (0, 0).
- Since point C is 20 miles east of point D, point C will be at (20, 0).
- Point B is 30 miles north of point C, so point B will be at (20, 30).
- Point A is 10 miles west of point B, so point A will be at (10, 30).

### Step 3: Determine the coordinates of points A and D
- Point A: (10, 30)
- Point D: (0, 0)

### Step 4: Use the distance formula to find the distance between A and D
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points A and D:
\[ d = \sqrt{(0 - 10)^2 + (0 - 30)^2} \]
\[ d = \sqrt{(-10)^2 + (-30)^2} \]
\[ d = \sqrt{100 + 900} \]
\[ d = \sqrt{1000} \]
\[ d = 10\sqrt{10} \]

### Conclusion
The distance between point A and point D is \( 10\sqrt{10} \) miles.

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