The direct distance between city A and city B is 350 miles. The direct distance between city B and city C is 525 miles. Which could be the direct distance between city C and city A? 75 miles 100 miles 175 miles

To determine the possible direct distance between city C and city A, we can use the **triangle inequality theorem**. This theorem states that for any triangle, the length of any one side must be: 1. **Less than the sum** of the other two sides. 2. **Greater than the difference** of the other two sides. Given: - Distance between **A and B (AB)** = 350 miles - Distance between **B and C (BC)** = 525 miles Let **CA** be the distance between city C and city A. According to the triangle inequality: 1. **CA < AB + BC** \[ CA < 350 + 525 \\ CA < 875 \text{ miles} \] 2. **CA > |AB - BC|** \[ CA > |350 - 525| \\ CA > 175 \text{ miles} \] So, the possible range for **CA** is: \[ 175 \text{ miles} < CA < 875 \text{ miles} \] **Evaluating the Options:** - **75 miles**: Less than 175 miles → **Not possible** - **100 miles**: Less than 175 miles → **Not possible** - **175 miles**: Exactly equal to 175 miles → **Possible** (if cities are collinear with C on the opposite side of B from A) **Conclusion:** The direct distance between city C and city A **could be 175 miles**. **Answer:** 175 miles