A fly is on the edge of a ceiling of a circular room with a radius of 58 feet. The fly walks straight across the ceiling to the opposite edge, passing through the center of the circle. It then walks straight to another point on the edge of the circle but not back through the center. The third part of the journey is straight back to the original starting point. If the third part of the journey was 80 feet long, how many total feet did the fly travel over the course of all three parts?
The fly's journey traces out the three sides of triangle.  Because one side of this triangle is a diameter of the circular ceiling and the other two sides are chords of this circle, the triangle is a right triangle. If the radius of the circular room is 58 feet, then the diameter is $2 \times 58 = 116$ feet. This is the hypotenuse of the right triangle. One of the legs is 80 feet, so the other leg must be equal to $\sqrt{116^2 - 80^2} = \sqrt{(13{,}456 - 6400)} = \sqrt{7056} = 84$ feet. The total distance traveled by the fly is $116 + 84 + 80 = \boxed{280}$ feet.
280
Level 4
Geometry
Problem: A fly is on the edge of a ceiling of a circular room with a radius of 58 feet. The fly walks straight across the ceiling to the opposite edge, passing through the center of the circle. It then walks straight to another point on the edge of the circle but not back through the center. The third part of the journey is straight back to the original starting point. If the third part of the journey was 80 feet long, how many total feet did the fly travel over the course of all three parts?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the total distance traveled by the fly, we need to calculate the distances of each part of its journey. 

1. The first part of the journey is from one edge of the circular room to the opposite edge, passing through the center. The diameter of the circle is twice the radius. Given the radius is 58 feet, the diameter is:
   Diameter = 2 * Radius = 2 * 58 = 116 feet.

2. The second part of the journey is to another point on the edge of the circle. We need to determine the distance from the point directly opposite the starting point to this new point. The fly's path forms a triangle with the center of the circle and the two points on the circumference. 

3. The third part of the journey is given as 80 feet, which is the distance from the new point back to the original starting point. 

4. To find the total distance traveled, we sum the distances of all three parts:
   Total Distance = Distance of first part + Distance of second part + Distance of third part.

5. We already calculated the first part as 116 feet. The second part's distance needs to be determined using the triangle formed. 

6. By applying the Law of Cosines or considering the geometry of the circle, we can find the distance of the second part. However, since we are not given the angle or specific coordinates, we can denote this distance as 'x'.

7. Therefore, the total distance can be expressed as:
   Total Distance = 116 + x + 80.

8. We know the total distance equals 280 feet, so we set up the equation:
   116 + x + 80 = 280.

9. Simplifying this gives:
   x + 196 = 280.

10. Solving for x yields:
    x = 280 - 196 = 84 feet.

11. Thus, the distance of the second part of the journey is 84 feet.

12. Finally, we can confirm the total distance traveled by the fly:
    Total Distance = 116 + 84 + 80 = 280 feet. 

The final answer is 280.<|END_OF_LATENT|> The final answer is $280$.