Question
Point A lies at (−4,4) and point B lies at (−3,6) . Which point is needed so that △ABC is scalene?(1 point) Responses C(−2,4) upper C left parenthesis negative 2 comma 4 right parenthesis C(−5,6) upper C left parenthesis negative 5 comma 6 right parenthesis C(−1,4) upper C left parenthesis negative 1 comma 4 right parenthesis C(−3,2)
Answers
GPT-4o mini
To create a scalene triangle \( \triangle ABC \), we need to identify a point C such that the lengths of all sides \( AB \), \( BC \), and \( CA \) are all different.
Given:
- Point A = (−4, 4)
- Point B = (−3, 6)
First, we calculate the length of side \( AB \):
\[
AB = \sqrt{((-3) - (-4))^2 + (6 - 4)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]
Now, let's evaluate the possible coordinates for point C:
1. **C(−2, 4)**:
- Length \( AC \):
\[
AC = \sqrt{((-2) - (-4))^2 + (4 - 4)^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2
\]
- Length \( BC \):
\[
BC = \sqrt{((-2) - (-3))^2 + (4 - 6)^2} = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]
- \( AC \) = 2, \( AB \) = \( \sqrt{5} \), \( BC \) = \( \sqrt{5} \) - not scalene.
2. **C(−5, 6)**:
- Length \( AC \):
\[
AC = \sqrt{((-5) - (-4))^2 + (6 - 4)^2} = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]
- Length \( BC \):
\[
BC = \sqrt{((-5) - (-3))^2 + (6 - 6)^2} = \sqrt{(-2)^2 + (0)^2} = \sqrt{4} = 2
\]
- \( AC \) = \( \sqrt{5} \), \( AB \) = \( \sqrt{5} \), \( BC \) = 2 - not scalene.
3. **C(−1, 4)**:
- Length \( AC \):
\[
AC = \sqrt{((-1) - (-4))^2 + (4 - 4)^2} = \sqrt{(3)^2 + (0)^2} = \sqrt{9} = 3
\]
- Length \( BC \):
\[
BC = \sqrt{((-1) - (-3))^2 + (4 - 6)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
- \( AC \) = 3, \( AB \) = \( \sqrt{5} \), \( BC = 2\sqrt{2} \) but still needs comparison of lengths: they are different - could be scalene.
4. **C(−3, 2)**:
- Length \( AC \):
\[
AC = \sqrt{((-3) - (-4))^2 + (2 - 4)^2} = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]
- Length \( BC \):
\[
BC = \sqrt{((-3) - (-3))^2 + (2 - 6)^2} = \sqrt{(0)^2 + (-4)^2} = \sqrt{16} = 4
\]
- \( AC = \sqrt{5} \), \( AB = \sqrt{5} \), \( BC = 4 \) - not scalene.
Therefore, we go back and summarize:
- **C(−1, 4)** has been identified as a candidate generating potential scalene configuration because \( AC \), \( AB \), and \( BC \) are distinct.
Thus, the correct choice that ensures that \( \triangle ABC \) is scalene is:
**C(−1, 4)**
Given:
- Point A = (−4, 4)
- Point B = (−3, 6)
First, we calculate the length of side \( AB \):
\[
AB = \sqrt{((-3) - (-4))^2 + (6 - 4)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]
Now, let's evaluate the possible coordinates for point C:
1. **C(−2, 4)**:
- Length \( AC \):
\[
AC = \sqrt{((-2) - (-4))^2 + (4 - 4)^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2
\]
- Length \( BC \):
\[
BC = \sqrt{((-2) - (-3))^2 + (4 - 6)^2} = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]
- \( AC \) = 2, \( AB \) = \( \sqrt{5} \), \( BC \) = \( \sqrt{5} \) - not scalene.
2. **C(−5, 6)**:
- Length \( AC \):
\[
AC = \sqrt{((-5) - (-4))^2 + (6 - 4)^2} = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]
- Length \( BC \):
\[
BC = \sqrt{((-5) - (-3))^2 + (6 - 6)^2} = \sqrt{(-2)^2 + (0)^2} = \sqrt{4} = 2
\]
- \( AC \) = \( \sqrt{5} \), \( AB \) = \( \sqrt{5} \), \( BC \) = 2 - not scalene.
3. **C(−1, 4)**:
- Length \( AC \):
\[
AC = \sqrt{((-1) - (-4))^2 + (4 - 4)^2} = \sqrt{(3)^2 + (0)^2} = \sqrt{9} = 3
\]
- Length \( BC \):
\[
BC = \sqrt{((-1) - (-3))^2 + (4 - 6)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
- \( AC \) = 3, \( AB \) = \( \sqrt{5} \), \( BC = 2\sqrt{2} \) but still needs comparison of lengths: they are different - could be scalene.
4. **C(−3, 2)**:
- Length \( AC \):
\[
AC = \sqrt{((-3) - (-4))^2 + (2 - 4)^2} = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]
- Length \( BC \):
\[
BC = \sqrt{((-3) - (-3))^2 + (2 - 6)^2} = \sqrt{(0)^2 + (-4)^2} = \sqrt{16} = 4
\]
- \( AC = \sqrt{5} \), \( AB = \sqrt{5} \), \( BC = 4 \) - not scalene.
Therefore, we go back and summarize:
- **C(−1, 4)** has been identified as a candidate generating potential scalene configuration because \( AC \), \( AB \), and \( BC \) are distinct.
Thus, the correct choice that ensures that \( \triangle ABC \) is scalene is:
**C(−1, 4)**