The graph shows the location of Gary's, Joan's, and Kash's houses on a coordinate plane. Each unit on the graph represents 1 mile.
A coordinate plane is shown. Joans house is located at 0, 6, Garys house is located at 5, 6, and Kashs house is located at 5, negative 2. Line segments connect the three points to form a right triangle.
Joan is going to pick up Gary and Kash for band practice at her house this afternoon. She is going to pick up Kash first, then Gary, and then go back home for practice. How many total miles will Joan travel while picking up her bandmates? Round your answer to the nearest mile. (4 points)
9 miles
10 miles
19 miles
22 miles
To find the total distance Joan will travel, we need to calculate the individual distances between her house and Kash's house, between Kash's house and Gary's house, and between Gary's house and Joan's house.
The distance between two points on a coordinate plane can be found using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using the given coordinates, we have:
- Distance between Joan's house (0, 6) and Kash's house (5, -2):
d1 = √((5 - 0)^2 + (-2 - 6)^2)
= √(5^2 + (-8)^2)
= √(25 + 64)
= √89
- Distance between Kash's house (5, -2) and Gary's house (5, 6):
d2 = √((5 - 5)^2 + (6 - (-2))^2)
= √(0^2 + 8^2)
= √(0 + 64)
= 8
- Distance between Gary's house (5, 6) and Joan's house (0, 6):
d3 = √((0 - 5)^2 + (6 - 6)^2)
= √((-5)^2 + 0^2)
= √(25 + 0)
= √25
= 5
Now, we can find the total distance Joan will travel by adding up these distances:
Total distance = d1 + d2 + d3
= √89 + 8 + 5
Approximating the square root and rounding the answer to the nearest mile, we get:
Total distance ≈ 9.43 + 8 + 5 ≈ 22
So, Joan will travel a total of 22 miles while picking up her bandmates.
The answer is 22 miles.