Question
Three boats in the ocean are located near a buoy as represented by the diagram below. The distance between boats 2 and 3 is known to be 65 miles.
Boat 1 is on the top and boat 2 is on the bottom right and boat 3 is on the bottom left. The boat 1, boat 2, and boat 3 are joined to form a triangle. A dotted line of length 30 miles from boat 1 meets the base of the triangle more than half way between boat 2 and boat 3 at Buoy. The distance between boat 2 and buoy is 25 miles.
Which statements are true?
Choose the two correct answers.
A.
The distance between boat 3 and the buoy is 40 miles.
B.
The distance between boats 1 and 2 is about 16.5 miles.
C.
The distance between boats 1 and 3 is 50 miles.
D.
The distance between the buoy and boat 1 is not needed to find the distance between boats 1 and 2.
E.
The distance between boats 2 and 3 is not needed to find the distance between the buoy and boat 3.
Boat 1 is on the top and boat 2 is on the bottom right and boat 3 is on the bottom left. The boat 1, boat 2, and boat 3 are joined to form a triangle. A dotted line of length 30 miles from boat 1 meets the base of the triangle more than half way between boat 2 and boat 3 at Buoy. The distance between boat 2 and buoy is 25 miles.
Which statements are true?
Choose the two correct answers.
A.
The distance between boat 3 and the buoy is 40 miles.
B.
The distance between boats 1 and 2 is about 16.5 miles.
C.
The distance between boats 1 and 3 is 50 miles.
D.
The distance between the buoy and boat 1 is not needed to find the distance between boats 1 and 2.
E.
The distance between boats 2 and 3 is not needed to find the distance between the buoy and boat 3.
Answers
GPT-4o mini
To analyze the statements, let's summarize the information we have:
- The distance between boats 2 and 3 is 65 miles.
- The buoy is located on the line segment between boats 2 and 3, 25 miles from boat 2.
- Therefore, the distance between boat 3 and the buoy is \(65 - 25 = 40\) miles.
- A dotted line of length 30 miles from boat 1 meets the line between boats 2 and 3 at the buoy.
Now, let's evaluate the statements:
A. **The distance between boat 3 and the buoy is 40 miles.**
- This statement is **TRUE**. We calculated that the distance between boat 3 and the buoy is 40 miles.
B. **The distance between boats 1 and 2 is about 16.5 miles.**
- We don't have enough information to confirm this distance with the data provided. This statement seems uncertain, so it is likely **FALSE**.
C. **The distance between boats 1 and 3 is 50 miles.**
- We cannot confirm this statement with the provided information. We also don't have the lengths of the sides of the triangle without additional data, so this statement is likely **FALSE**.
D. **The distance between the buoy and boat 1 is not needed to find the distance between boats 1 and 2.**
- This statement is **TRUE** because we can find the distance between boats 1 and 2 using the other given distances without needing the distance from boat 1 to the buoy.
E. **The distance between boats 2 and 3 is not needed to find the distance between the buoy and boat 3.**
- This statement is **FALSE** because we specifically need the distance between boats 2 and 3 to find the distance from buoy to boat 3, which we calculated based on the distance from boat 2 to the buoy.
Based on the evaluation, the two correct statements are:
- A. The distance between boat 3 and the buoy is 40 miles.
- D. The distance between the buoy and boat 1 is not needed to find the distance between boats 1 and 2.
- The distance between boats 2 and 3 is 65 miles.
- The buoy is located on the line segment between boats 2 and 3, 25 miles from boat 2.
- Therefore, the distance between boat 3 and the buoy is \(65 - 25 = 40\) miles.
- A dotted line of length 30 miles from boat 1 meets the line between boats 2 and 3 at the buoy.
Now, let's evaluate the statements:
A. **The distance between boat 3 and the buoy is 40 miles.**
- This statement is **TRUE**. We calculated that the distance between boat 3 and the buoy is 40 miles.
B. **The distance between boats 1 and 2 is about 16.5 miles.**
- We don't have enough information to confirm this distance with the data provided. This statement seems uncertain, so it is likely **FALSE**.
C. **The distance between boats 1 and 3 is 50 miles.**
- We cannot confirm this statement with the provided information. We also don't have the lengths of the sides of the triangle without additional data, so this statement is likely **FALSE**.
D. **The distance between the buoy and boat 1 is not needed to find the distance between boats 1 and 2.**
- This statement is **TRUE** because we can find the distance between boats 1 and 2 using the other given distances without needing the distance from boat 1 to the buoy.
E. **The distance between boats 2 and 3 is not needed to find the distance between the buoy and boat 3.**
- This statement is **FALSE** because we specifically need the distance between boats 2 and 3 to find the distance from buoy to boat 3, which we calculated based on the distance from boat 2 to the buoy.
Based on the evaluation, the two correct statements are:
- A. The distance between boat 3 and the buoy is 40 miles.
- D. The distance between the buoy and boat 1 is not needed to find the distance between boats 1 and 2.