Question

Elena's office is 25 miles southeast of her home. Which of the following correctly uses a Pythagorean triple to find out how many miles Elena needs to drive south and then east to get from home to work? (1 point)
• 7 miles south and 18 miles east
• 5 miles south and 24.5 miles east
• 7 miles south and 24 miles east
• 5 miles south and 20 miles east

Answer 1

To determine which option accurately uses a Pythagorean triple to represent Elena's 25-mile diagonal distance from home to her office, we need to check whether the given combinations of south and east distances satisfy the Pythagorean theorem, \( a^2 + b^2 = c^2 \), where \( c \) is the distance from home to office.

Given that her office is 25 miles away, we have \( c = 25 \).

Let's evaluate each option:

1. **7 miles south and 18 miles east:**
\[
a^2 + b^2 = 7^2 + 18^2 = 49 + 324 = 373
\]
Since \( \sqrt{373} \) is not 25, this option is incorrect.

2. **5 miles south and 24.5 miles east:**
\[
a^2 + b^2 = 5^2 + 24.5^2 = 25 + 600.25 = 625.25
\]
Since \( \sqrt{625.25} \) is not 25, this option is incorrect.

3. **7 miles south and 24 miles east:**
\[
a^2 + b^2 = 7^2 + 24^2 = 49 + 576 = 625
\]
Since \( \sqrt{625} = 25 \), this option is correct.

4. **5 miles south and 20 miles east:**
\[
a^2 + b^2 = 5^2 + 20^2 = 25 + 400 = 425
\]
Since \( \sqrt{425} \) is not 25, this option is incorrect.

From the options given, the correct choice that uses a Pythagorean triple is:
**7 miles south and 24 miles east.**