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(−3,2)
cap c times open paren negative 3 comma 2 close paren - no response given

C(−5,6)
cap c times open paren negative 5 comma 6 close paren - no response given

C(−1,4)
cap c times open paren negative 1 comma 4 close paren - no response given

C(−2,4)

Answer 1

To determine which point \( C \) will make triangle \( ABC \) scalene, we need to ensure that all sides of triangle \( ABC \) have different lengths.

Points \( A \) and \( B \) are given as:
- \( A(-4, 4) \)
- \( B(-3, 6) \)

We can calculate the distance \( AB \) between A and B using the distance formula:
\[
AB = \sqrt{(-3 - (-4))^2 + (6 - 4)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]

Next, we will check each candidate point \( C \) and calculate distances \( AC \) and \( BC \) to see if they all differ.

### Checking Candidate Points:

1. \( C(-3, 2) \)
- \( AC = \sqrt{(-3 - (-4))^2 + (2 - 4)^2} = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \)
- \( BC = \sqrt{(-3 - (-3))^2 + (2 - 6)^2} = \sqrt{(0)^2 + (-4)^2} = \sqrt{16} = 4 \)
- Sides: \( AB = \sqrt{5}, AC = \sqrt{5}, BC = 4 \) (Not scalene, \( AC = AB \))

2. \( C(-5, 6) \)
- \( AC = \sqrt{(-5 - (-4))^2 + (6 - 4)^2} = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \)
- \( BC = \sqrt{(-5 - (-3))^2 + (6 - 6)^2} = \sqrt{(-2)^2 + (0)^2} = \sqrt{4} = 2 \)
- Sides: \( AB = \sqrt{5}, AC = \sqrt{5}, BC = 2 \) (Not scalene, \( AC = AB \))

3. \( C(-1, 4) \)
- \( AC = \sqrt{(-1 - (-4))^2 + (4 - 4)^2} = \sqrt{(3)^2 + (0)^2} = 3 \)
- \( BC = \sqrt{(-1 - (-3))^2 + (4 - 6)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \)
- Sides: \( AB = \sqrt{5}, AC = 3, BC = 2\sqrt{2} \) (Different lengths, scalene)

4. \( C(-2, 4) \)
- \( AC = \sqrt{(-2 - (-4))^2 + (4 - 4)^2} = \sqrt{(2)^2 + (0)^2} = 2 \)
- \( BC = \sqrt{(-2 - (-3))^2 + (4 - 6)^2} = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \)
- Sides: \( AB = \sqrt{5}, AC = 2, BC = \sqrt{5} \) (Not scalene, \( BC = AB \))

### Conclusion:
The only point \( C \) that results in all sides of triangle \( ABC \) being of different lengths (making it scalene) is:
**C(-1, 4)**.