Question

Point A lies at (−4,4) and point B lies at (−3,6) . Which point is needed so that △ABC is scalene (all 3 sides have different length)?

(1 point)
Responses

C(−2,4)cap c times open paren negative 2 comma 4 close paren

C(−3,2)cap c times open paren negative 3 comma 2 close paren

C(−5,6)cap c times open paren negative 5 comma 6 close paren

C(−1,4)cap c times open paren negative 1 comma 4 close paren

Answer 1

To determine which point \( C \) makes triangle \( \triangle ABC \) scalene, we need to check the lengths of the sides formed by the points \( A(-4, 4) \), \( B(-3, 6) \), and each candidate for point \( C \).

1. **Calculate the length of \( AB \)**:
\[
AB = \sqrt{((-4 - (-3))^2 + (4 - 6)^2)} = \sqrt{((-1)^2 + (-2)^2)} = \sqrt{1 + 4} = \sqrt{5}
\]

2. **Evaluate each candidate point \( C \)**:

- **For \( C(-2,4) \)**:
\[
AC = \sqrt{((-4 - (-2))^2 + (4 - 4)^2)} = \sqrt{((-2)^2 + 0^2)} = \sqrt{4} = 2
\]
\[
BC = \sqrt{((-3 - (-2))^2 + (6 - 4)^2)} = \sqrt{((-1)^2 + (2)^2)} = \sqrt{1 + 4} = \sqrt{5}
\]
- **Side lengths**: \( AB = \sqrt{5}, AC = 2, BC = \sqrt{5} \)
- **Not scalene** (two sides are equal).

- **For \( C(-3,2) \)**:
\[
AC = \sqrt{((-4 - (-3))^2 + (4 - 2)^2)} = \sqrt{((-1)^2 + (2)^2)} = \sqrt{1 + 4} = \sqrt{5}
\]
\[
BC = \sqrt{((-3 - (-3))^2 + (6 - 2)^2)} = \sqrt{(0^2 + (4)^2)} = \sqrt{16} = 4
\]
- **Side lengths**: \( AB = \sqrt{5}, AC = \sqrt{5}, BC = 4 \)
- **Not scalene** (two sides are equal).

- **For \( C(-5,6) \)**:
\[
AC = \sqrt{((-4 - (-5))^2 + (4 - 6)^2)} = \sqrt{((1)^2 + (-2)^2)} = \sqrt{1 + 4} = \sqrt{5}
\]
\[
BC = \sqrt{((-3 - (-5))^2 + (6 - 6)^2)} = \sqrt{((2)^2 + (0)^2)} = \sqrt{4} = 2
\]
- **Side lengths**: \( AB = \sqrt{5}, AC = \sqrt{5}, BC = 2 \)
- **Not scalene** (two sides are equal).

- **For \( C(-1,4) \)**:
\[
AC = \sqrt{((-4 - (-1))^2 + (4 - 4)^2)} = \sqrt{((-3)^2 + 0^2)} = \sqrt{9} = 3
\]
\[
BC = \sqrt{((-3 - (-1))^2 + (6 - 4)^2)} = \sqrt{((-2)^2 + (2)^2)} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
- **Side lengths**: \( AB = \sqrt{5}, AC = 3, BC = 2\sqrt{2} \)
- Since \( \sqrt{5} \), \( 3 \), and \( 2\sqrt{2} \) are all different, \( \triangle ABC \) is **scalene**.

Thus, the point needed for \( \triangle ABC \) to be scalene is:

\( C(-1, 4) \)