Two bicyclists, starting at the same place, are riding toward the same compground by two different routes. One cyclist rides 1080 m due east and then turns due north and travels norther 1430 m before reaching the campground. The second cyclist starts out by heading due north for 1950 m and then turns and heads directly toward the compground. At the turning point, how far is the second cyclist from the campground ?

Options

A. 1200 m
B. 1700m
C. 1100 m
D. 1600 m

Answer

To solve the problem step by step, we will break down the movements of both cyclists and use the Pythagorean theorem to find the distance from the second cyclist to the campground at the turning point.

### Step 1: Understand the movements of the cyclists
- **Cyclist 1 (C1)** travels:
- 1080 m due east
- Then turns north and travels 1430 m to reach the campground.

- **Cyclist 2 (C2)** travels:
- 1950 m due north
- Then turns directly towards the campground.

### Step 2: Create a diagram
- Start by marking the starting point (O).
- For Cyclist 1:
- From point O, move 1080 m east to point A.
- From point A, move 1430 m north to point B (the campground).

- For Cyclist 2:
- From point O, move 1950 m north to point D (the turning point).

### Step 3: Determine the coordinates
- **Coordinates for Cyclist 1 (C1)**:
- Point A (1080, 0)
- Point B (1080, 1430)

- **Coordinates for Cyclist 2 (C2)**:
- Point D (0, 1950)

### Step 4: Calculate the distance from point D to point B (the campground)
- The coordinates of point B (the campground) are (1080, 1430).
- The coordinates of point D (the turning point for C2) are (0, 1950).

### Step 5: Use the distance formula
To find the distance \( x \) from point D to point B, we can use the distance formula:
\[ x = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Where:
- \( (x_1, y_1) \) are the coordinates of point D (0, 1950)
- \( (x_2, y_2) \) are the coordinates of point B (1080, 1430)

Substituting the values:
\[ x = \sqrt{(1080 - 0)^2 + (1430 - 1950)^2} \]
\[ x = \sqrt{(1080)^2 + (-520)^2} \]
\[ x = \sqrt{1166400 + 270400} \]
\[ x = \sqrt{1436800} \]
\[ x \approx 1198.66 \text{ m} \]

### Step 6: Round the answer
The distance from the second cyclist to the campground at the turning point is approximately 1198.66 m, which can be rounded to 1200 m.

### Final Answer
The second cyclist is approximately **1200 m** from the campground at the turning point.

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