Question
Part C: Find the shortest distance, in miles, from Math Middle School to Hypotenuse High School. Show every step of your work. (4 points)
Part D: Deon traveled from Hypotenuse High to Euclid Elementary and then to Math Middle. Kaya traveled from Hypotenuse High to Math Middle along a straight path. Who went the shortest distance? Explain. (4 points)
Part D: Deon traveled from Hypotenuse High to Euclid Elementary and then to Math Middle. Kaya traveled from Hypotenuse High to Math Middle along a straight path. Who went the shortest distance? Explain. (4 points)
Answers
GPT-4o mini
To solve Part C and D, let's assume we have the relevant coordinates for each school and then analyze each step.
### Part C: Find the Shortest Distance
1. **Identify the Coordinates:**
Assume:
- Math Middle School (M) is at point (x1, y1).
- Hypotenuse High School (H) is at point (x2, y2).
Let's use hypothetical coordinates for illustration:
- M: (2, 3)
- H: (8, 7)
2. **Calculate the Distance Using the Distance Formula:**
The distance \( d \) between two points \((x1, y1)\) and \((x2, y2)\) is calculated using the formula:
\[
d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}
\]
Plugging in our hypothetical values:
\[
d = \sqrt{(8 - 2)^2 + (7 - 3)^2}
\]
\[
d = \sqrt{(6)^2 + (4)^2}
\]
\[
d = \sqrt{36 + 16}
\]
\[
d = \sqrt{52}
\]
\[
d \approx 7.21 \text{ miles}
\]
Therefore, the distance from Math Middle School to Hypotenuse High School is approximately 7.21 miles.
### Part D: Analyze Travel Distances
1. **Distance Traveled by Deon:**
- Deon goes from Hypotenuse High (H) to Euclid Elementary (E) and then to Math Middle (M).
- Assume Euclid Elementary is at (x3, y3). For example, let’s say E: (5, 5).
- First, calculate the distance from H to E:
\[
d_{HE} = \sqrt{(5 - 8)^2 + (5 - 7)^2}
\]
\[
= \sqrt{(-3)^2 + (-2)^2}
\]
\[
= \sqrt{9 + 4}
\]
\[
= \sqrt{13} \approx 3.61 \text{ miles}
\]
- Then calculate the distance from E to M:
\[
d_{EM} = \sqrt{(2 - 5)^2 + (3 - 5)^2}
\]
\[
= \sqrt{(-3)^2 + (-2)^2}
\]
\[
= \sqrt{9 + 4}
\]
\[
= \sqrt{13} \approx 3.61 \text{ miles}
\]
- Total distance traveled by Deon:
\[
d_{Deon} = d_{HE} + d_{EM} \approx 3.61 + 3.61 \approx 7.22 \text{ miles}
\]
2. **Distance Traveled by Kaya:**
- Kaya travels directly from H to M, which we already calculated as approximately 7.21 miles.
3. **Comparison:**
- Deon's total distance is approximately: 7.22 miles.
- Kaya's distance is approximately: 7.21 miles.
### Conclusion:
Kaya went the shortest distance, as her total distance of approximately **7.21 miles** is less than Deon's total distance of approximately **7.22 miles**. Therefore, Kaya took the more direct route straight from Hypotenuse High to Math Middle, while Deon took a longer indirect path.
### Part C: Find the Shortest Distance
1. **Identify the Coordinates:**
Assume:
- Math Middle School (M) is at point (x1, y1).
- Hypotenuse High School (H) is at point (x2, y2).
Let's use hypothetical coordinates for illustration:
- M: (2, 3)
- H: (8, 7)
2. **Calculate the Distance Using the Distance Formula:**
The distance \( d \) between two points \((x1, y1)\) and \((x2, y2)\) is calculated using the formula:
\[
d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}
\]
Plugging in our hypothetical values:
\[
d = \sqrt{(8 - 2)^2 + (7 - 3)^2}
\]
\[
d = \sqrt{(6)^2 + (4)^2}
\]
\[
d = \sqrt{36 + 16}
\]
\[
d = \sqrt{52}
\]
\[
d \approx 7.21 \text{ miles}
\]
Therefore, the distance from Math Middle School to Hypotenuse High School is approximately 7.21 miles.
### Part D: Analyze Travel Distances
1. **Distance Traveled by Deon:**
- Deon goes from Hypotenuse High (H) to Euclid Elementary (E) and then to Math Middle (M).
- Assume Euclid Elementary is at (x3, y3). For example, let’s say E: (5, 5).
- First, calculate the distance from H to E:
\[
d_{HE} = \sqrt{(5 - 8)^2 + (5 - 7)^2}
\]
\[
= \sqrt{(-3)^2 + (-2)^2}
\]
\[
= \sqrt{9 + 4}
\]
\[
= \sqrt{13} \approx 3.61 \text{ miles}
\]
- Then calculate the distance from E to M:
\[
d_{EM} = \sqrt{(2 - 5)^2 + (3 - 5)^2}
\]
\[
= \sqrt{(-3)^2 + (-2)^2}
\]
\[
= \sqrt{9 + 4}
\]
\[
= \sqrt{13} \approx 3.61 \text{ miles}
\]
- Total distance traveled by Deon:
\[
d_{Deon} = d_{HE} + d_{EM} \approx 3.61 + 3.61 \approx 7.22 \text{ miles}
\]
2. **Distance Traveled by Kaya:**
- Kaya travels directly from H to M, which we already calculated as approximately 7.21 miles.
3. **Comparison:**
- Deon's total distance is approximately: 7.22 miles.
- Kaya's distance is approximately: 7.21 miles.
### Conclusion:
Kaya went the shortest distance, as her total distance of approximately **7.21 miles** is less than Deon's total distance of approximately **7.22 miles**. Therefore, Kaya took the more direct route straight from Hypotenuse High to Math Middle, while Deon took a longer indirect path.