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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

1 answer

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: \( 3 \)