Abigail lives at the point (-3, -1) and wants to travel to her friend Clara's house at (2, 3) on the map. She has two choices to get there: either travel directly to Clara's house, or Abigail can stop by Bob's Store at (2, -1) to pick up snacks before travelling to Clara's house. Abigail wants to compare the two distances of her choices
To compare the two distances, we can use the distance formula.
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
If Abigail travels directly to Clara's house, the distance between her house and Clara's house would be:
d1 = √((2 - (-3))^2 + (3 - (-1))^2)
= √(5^2 + 4^2)
= √(25 + 16)
= √41
If Abigail stops by Bob's Store before going to Clara's house, she would have to travel from her house to Bob's Store and then from Bob's Store to Clara's house. The total distance would be the sum of the distance from her house to Bob's Store and the distance from Bob's Store to Clara's house.
The distance from her house to Bob's Store would be:
d2a = √((2 - (-3))^2 + (-1 - (-1))^2)
= √(5^2 + 0^2)
= √(25 + 0)
= 5
The distance from Bob's Store to Clara's house would be:
d2b = √((2 - 2)^2 + (3 - (-1))^2)
= √(0^2 + 4^2)
= √(0 + 16)
= 4
Therefore, the total distance if Abigail stops by Bob's Store would be:
d2 = d2a + d2b
= 5 + 4
= 9
Comparing the two distances, we see that √41 (approximately 6.4) is smaller than 9. Therefore, the direct distance to Clara's house is shorter than the distance if Abigail stops by Bob's Store.
To compare the two distances, we need to calculate the distance from Abigail's house to Clara's house, as well as the distance from Abigail's house to Bob's Store and then from Bob's Store to Clara's house.
To calculate the distance between two points (x1, y1) and (x2, y2) on a coordinate plane, you can use the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the distances:
1. Distance from Abigail's house to Clara's house:
x1 = -3, y1 = -1 (Abigail's house)
x2 = 2, y2 = 3 (Clara's house)
d1 = sqrt((2 - (-3))^2 + (3 - (-1))^2)
= sqrt(5^2 + 4^2)
= sqrt(25 + 16)
= sqrt(41)
2. Distance from Abigail's house to Bob's Store:
x1 = -3, y1 = -1 (Abigail's house)
x2 = 2, y2 = -1 (Bob's Store)
d2 = sqrt((2 - (-3))^2 + (-1 - (-1))^2)
= sqrt(5^2 + 0^2)
= sqrt(25 + 0)
= sqrt(25)
= 5
3. Distance from Bob's Store to Clara's house:
x1 = 2, y1 = -1 (Bob's Store)
x2 = 2, y2 = 3 (Clara's house)
d3 = sqrt((2 - 2)^2 + (3 - (-1))^2)
= sqrt(0^2 + 4^2)
= sqrt(0 + 16)
= sqrt(16)
= 4
Now we compare the two distances:
Option 1: Travel directly to Clara's house.
Distance = sqrt(41)
Option 2: Stop by Bob's Store and then travel to Clara's house.
Distance = d2 + d3
= 5 + 4
= 9
Comparing the two distances, we can see that traveling directly to Clara's house is shorter (sqrt(41) < 9). Therefore, Abigail should choose to travel directly to Clara's house.
To compare the two distances, we need to calculate the distance of each option.
Option 1: Directly to Clara's house.
We can use the distance formula to find the distance between two points in a coordinate plane:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using Abigail's coordinates (-3, -1) and Clara's house coordinates (2, 3), we can calculate the distance as follows:
d1 = √((2 - (-3))^2 + (3 - (-1))^2)
= √((2 + 3)^2 + (3 + 1)^2)
= √(5^2 + 4^2)
= √(25 + 16)
= √41
So the distance directly to Clara's house is √41.
Option 2: Stop by Bob's Store first.
Using Abigail's coordinates (-3, -1) and Bob's Store coordinates (2, -1), we can calculate the distance as follows:
d2 = √((2 - (-3))^2 + (-1 - (-1))^2)
= √((2 + 3)^2 + (0)^2)
= √(5^2 + 0)
= √25
= 5
So the distance from Abigail's house to Bob's Store is 5 units.
Next, we need to calculate the distance from Bob's Store to Clara's house:
d3 = √((2 - 2)^2 + (3 - (-1))^2)
= √(0^2 + 4^2)
= √16
= 4
Therefore, the distance from Bob's Store to Clara's house is 4 units.
Now, we can find the total distance of Option 2 (stop by Bob's Store first) by adding the distances:
Total distance = distance from Abigail's house to Bob's Store + distance from Bob's Store to Clara's house
= 5 + 4
= 9
So the total distance of Option 2 is 9 units.
Comparing the distances:
- The distance of Option 1 (directly to Clara's house) is √41.
- The distance of Option 2 (stop by Bob's Store first) is 9 units.
Therefore, Abigail's best option for the shortest distance is to travel directly to Clara's house.