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(−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(−2,4)
upper C left parenthesis negative 2 comma 4 right parenthesis

C(−3,2)

Answer 1

To determine which point C will make triangle ABC scalene, we need to find a point C such that all three sides of triangle ABC have different lengths.

First, we calculate the lengths of the sides AB, AC, and BC for each point C provided in the options:

1. **Length AB** (between points A(-4,4) and B(-3,6)):
\[
AB = \sqrt{((-3) - (-4))^2 + (6 - 4)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]

Now let’s calculate the lengths for each point C given:

### Option 1: 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:
\[
BC = \sqrt{((-5) - (-3))^2 + (6 - 6)^2} = \sqrt{(-2)^2 + (0)^2} = \sqrt{4} = 2
\]

Resulting lengths: AB = \(\sqrt{5}\), AC = \(\sqrt{5}\), BC = 2 (not scalene).

### Option 2: C(−1,4)
- Length AC:
\[
AC = \sqrt{((-1) - (-4))^2 + (4 - 4)^2} = \sqrt{(3)^2 + (0)^2} = 3
\]

- Length BC:
\[
BC = \sqrt{((-1) - (-3))^2 + (4 - 6)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

Resulting lengths: AB = \(\sqrt{5}\), AC = 3, BC = \(2\sqrt{2}\) (all different, scalene).

### Option 3: C(−2,4)
- Length AC:
\[
AC = \sqrt{((-2) - (-4))^2 + (4 - 4)^2} = \sqrt{(2)^2 + (0)^2} = 2
\]

- Length BC:
\[
BC = \sqrt{((-2) - (-3))^2 + (4 - 6)^2} = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]

Resulting lengths: AB = \(\sqrt{5}\), AC = 2, BC = \(\sqrt{5}\) (not scalene).

### Option 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:
\[
BC = \sqrt{((-3) - (-3))^2 + (2 - 6)^2} = \sqrt{(0)^2 + (-4)^2} = 4
\]

Resulting lengths: AB = \(\sqrt{5}\), AC = \(\sqrt{5}\), BC = 4 (not scalene).

### Conclusion:
The only option that results in all sides having different lengths is **C(−1,4)**. Therefore, the answer is:

**C(−1,4)** (scalene triangle).