Question
Use a coordinate proof to show that triangle ABC is scalene. What are the lengths of sides AB , BC , and AC ?
(1 point)
Responses
AB≈10.6, BC≈8.1, and AC≈7.1; therefore, ABC is a scalene triangle.
cap A cap b almost equal to 10 point 6 , cap b cap c almost equal to 8 point 1 , and cap A cap c almost equal to 7 point 1 ; therefore, upper A upper B upper C is a scalene triangle.
AB≈8.5, BC≈6.4, and AC≈7.1; therefore, ABC is a scalene triangle.
cap A cap b almost equal to 8 point 5 , cap b cap c almost equal to 6 point 4 , and cap A cap c almost equal to 7 point 1 ; therefore, upper A upper B upper C is a scalene triangle.
AB≈3.6, BC≈6.1, and AC≈4.2; therefore, ABC is a scalene triangle.
cap A cap b almost equal to 3 point 6 , cap b cap c almost equal to 6 point 1 , and cap A cap c almost equal to 4 point 2 ; therefore, upper A upper B upper C is a scalene triangle.
AB≈10.6, BC≈6.4, and AC≈7.1; therefore, ABC is a scalene triangle.
(1 point)
Responses
AB≈10.6, BC≈8.1, and AC≈7.1; therefore, ABC is a scalene triangle.
cap A cap b almost equal to 10 point 6 , cap b cap c almost equal to 8 point 1 , and cap A cap c almost equal to 7 point 1 ; therefore, upper A upper B upper C is a scalene triangle.
AB≈8.5, BC≈6.4, and AC≈7.1; therefore, ABC is a scalene triangle.
cap A cap b almost equal to 8 point 5 , cap b cap c almost equal to 6 point 4 , and cap A cap c almost equal to 7 point 1 ; therefore, upper A upper B upper C is a scalene triangle.
AB≈3.6, BC≈6.1, and AC≈4.2; therefore, ABC is a scalene triangle.
cap A cap b almost equal to 3 point 6 , cap b cap c almost equal to 6 point 1 , and cap A cap c almost equal to 4 point 2 ; therefore, upper A upper B upper C is a scalene triangle.
AB≈10.6, BC≈6.4, and AC≈7.1; therefore, ABC is a scalene triangle.
Answers
GPT-4o mini
To demonstrate that triangle ABC is scalene using a coordinate proof, we need to show that all the sides have different lengths. We can use the distance formula to find the lengths of sides \( AB \), \( BC \), and \( AC \).
The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Let's assume we have the coordinates of points A, B, and C:
- Let \( A = (x_1, y_1) \)
- Let \( B = (x_2, y_2) \)
- Let \( C = (x_3, y_3) \)
We can compute each side's length as follows:
1. **Length of \( AB \)**:
\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
2. **Length of \( BC \)**:
\[
BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}
\]
3. **Length of \( AC \)**:
\[
AC = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2}
\]
After calculating the lengths \( AB \), \( BC \), and \( AC \), if we find that all three lengths are different, we can conclude that triangle ABC is scalene.
Now, let's check the provided answer options for lengths.
1. From the first response, we have:
- \( AB \approx 10.6 \), \( BC \approx 8.1 \), \( AC \approx 7.1 \)
2. From the second response, we have:
- \( AB \approx 8.5 \), \( BC \approx 6.4 \), \( AC \approx 7.1 \)
3. From the third response:
- \( AB \approx 3.6 \), \( BC \approx 6.1 \), \( AC \approx 4.2 \)
4. From the fourth response:
- \( AB \approx 10.6 \), \( BC \approx 6.4 \), \( AC \approx 7.1 \)
For triangle ABC to be scalene, all these values must be different.
Among the provided options, the first response \( (AB \approx 10.6, BC \approx 8.1, AC \approx 7.1) \) shows all side lengths are different. Therefore, we can conclude:
**Answer**: Using the lengths of \( AB \approx 10.6 \), \( BC \approx 8.1 \), and \( AC \approx 7.1 \); therefore, triangle ABC is a scalene triangle.
The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Let's assume we have the coordinates of points A, B, and C:
- Let \( A = (x_1, y_1) \)
- Let \( B = (x_2, y_2) \)
- Let \( C = (x_3, y_3) \)
We can compute each side's length as follows:
1. **Length of \( AB \)**:
\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
2. **Length of \( BC \)**:
\[
BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}
\]
3. **Length of \( AC \)**:
\[
AC = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2}
\]
After calculating the lengths \( AB \), \( BC \), and \( AC \), if we find that all three lengths are different, we can conclude that triangle ABC is scalene.
Now, let's check the provided answer options for lengths.
1. From the first response, we have:
- \( AB \approx 10.6 \), \( BC \approx 8.1 \), \( AC \approx 7.1 \)
2. From the second response, we have:
- \( AB \approx 8.5 \), \( BC \approx 6.4 \), \( AC \approx 7.1 \)
3. From the third response:
- \( AB \approx 3.6 \), \( BC \approx 6.1 \), \( AC \approx 4.2 \)
4. From the fourth response:
- \( AB \approx 10.6 \), \( BC \approx 6.4 \), \( AC \approx 7.1 \)
For triangle ABC to be scalene, all these values must be different.
Among the provided options, the first response \( (AB \approx 10.6, BC \approx 8.1, AC \approx 7.1) \) shows all side lengths are different. Therefore, we can conclude:
**Answer**: Using the lengths of \( AB \approx 10.6 \), \( BC \approx 8.1 \), and \( AC \approx 7.1 \); therefore, triangle ABC is a scalene triangle.