Question

9. Which of the following is an equation of a line that is perpendicular to y = -2x + 5 and passes through the point (-2, 3)?
A) y = −12x + 2
B) y = -2x - 1
C) y = 2x + 7
D) y = 12x + 4

11. Find the missing side lengths, perimeter, and area of triangle ABC on the coordinate plane. All answers should be in simplest radical form, if necessary: Note: the formula for area of a triangle is A = bh/2, where b is the length of the base (BC), and h is the height from side BC to point A.
Point A - (-1, 4)
Point B - (-3, 2)
Point C - (0, 2)
Side Length AB: _______
Side Length BC: _______
Side Length AC: _______
Perimeter of Triangle ABC: _______
Area of Triangle ABC: _______
A) 7 + Sq8
B) 2
C) Sq8
D) Sq5
E) 5
F) 3
G) 2Sq2
H) 3 + 2Sq + Sq5
I) Sq3

Answer 1

Let's tackle each question separately.

### Question 9: Find the equation of a line perpendicular to \( y = -2x + 5 \) and passing through the point (-2, 3).

1. **Find the slope of the line given**:
The slope (m) of the line \( y = -2x + 5 \) is -2.

2. **Find the slope of the perpendicular line**:
The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. Therefore:
\[
m_{\text{perpendicular}} = -\frac{1}{-2} = \frac{1}{2}
\]

3. **Use the point-slope form of the equation**:
The point-slope form of a line is \( y - y_1 = m(x - x_1) \).
Using the point (-2, 3):
\[
y - 3 = \frac{1}{2}(x + 2)
\]
Simplifying:
\[
y - 3 = \frac{1}{2}x + 1 \\
y = \frac{1}{2}x + 4
\]

Now we need to check which option corresponds to this form. Converting to standard form:
Multiply through by 2:
\[
2y = x + 8 \\
x - 2y + 8 = 0 \\
y = \frac{1}{2}x + 4 \implies \text{is equivalent to } y = 2x - 4
\]
However, this is not in the given options. There might be a need to recheck against the provided options for a reinterpretation.

### Question 11: Find the side lengths, perimeter, and area of triangle ABC.

1. **Find the lengths of the sides**:
- **Length AB**:
\[
A(-1, 4), B(-3, 2) \\
AB = \sqrt{((-1) - (-3))^2 + (4 - 2)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

- **Length BC**:
\[
B(-3, 2), C(0, 2) \\
BC = \sqrt{((-3) - (0))^2 + (2 - 2)^2} = \sqrt{(-3)^2 + 0^2} = \sqrt{9} = 3
\]

- **Length AC**:
\[
A(-1, 4), C(0, 2) \\
AC = \sqrt{((-1) - (0))^2 + (4 - 2)^2} = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]

2. **Perimeter**:
\[
\text{Perimeter} = AB + BC + AC = 2\sqrt{2} + 3 + \sqrt{5}
\]

3. **Area** (find height from A perpendicular to BC):
BC is a horizontal line (y = 2).
The height from A to line BC is the vertical distance from A(−1, 4) to y = 2:
Height \( h = 4 - 2 = 2 \)

Using base \( b = BC = 3 \):
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 2 = 3
\]

### Results for Question 11:
- **Side Lengths**:
- AB: \( 2\sqrt{2} \)
- BC: \( 3 \)
- AC: \( \sqrt{5} \)

- **Perimeter**:
- \( 2\sqrt{2} + 3 + \sqrt{5} \)

- **Area**:
- \( 3 \)

### Final Answers Recap:
1. For Question 9: The resulting equation isn't directly listed in your options; however, the calculations suggest it should be checked further.

2. For Question 11:
- Side Length AB: \( 2\sqrt{2} \)
- Side Length BC: \( 3 \)
- Side Length AC: \( \sqrt{5} \)
- Perimeter: \( 2\sqrt{2} + 3 + \sqrt{5} \)
- Area: \( 3 \)