Question

Sanjay, Colby, and Arpitha are training to run a marathon. On Saturday, Sanjay ran StartRoot 125 EndRoot miles, Colby ran 13 miles, and Arpitha ran the shortest route from the library to her house, as shown in the diagram.

A right triangle. The distance from the angle with measure 90 degrees to Arpitha's Home is 9 miles, and the distance to the Library is 6 miles.

Which list shows the names in order from the person who ran the shortest distance to the one who ran the greatest distance?
Colby, Sanjay, Arpitha
Sanjay, Arpitha, Colby
Sanjay, Colby, Arpitha
Arpitha, Sanjay, Colby

Answer 1

To find the distances each person ran, we need to calculate Arpitha's distance using the Pythagorean theorem because she ran along the sides of a right triangle.

We have:
- One leg of the triangle (distance from the angle with measure 90 degrees to Arpitha's Home) = 9 miles
- The other leg of the triangle (distance to the Library) = 6 miles

Using the Pythagorean theorem:

\[
\text{Distance}^2 = 9^2 + 6^2
\]
\[
\text{Distance}^2 = 81 + 36
\]
\[
\text{Distance}^2 = 117
\]
\[
\text{Distance} = \sqrt{117}
\]

Now let's calculate the distances for everyone:

1. **Sanjay** ran \( \sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5} \approx 11.18 \) miles.
2. **Colby** ran 13 miles.
3. **Arpitha** ran \( \sqrt{117} \approx 10.82 \) miles.

Now, let's compare the distances:
- Arpitha: \( \approx 10.82 \) miles
- Sanjay: \( \approx 11.18 \) miles
- Colby: 13 miles

In order from the shortest to the longest distance:
1. Arpitha (approximately 10.82 miles)
2. Sanjay (approximately 11.18 miles)
3. Colby (13 miles)

Thus, the correct order is **Arpitha, Sanjay, Colby**.

So the answer is:
**Arpitha, Sanjay, Colby.**