Let's find the perimeter of the rectangle ABCD through a step-by-step approach:
1. Identify the vertices and their coordinates:
- Vertex A: [tex]\((-1, 9)\)[/tex]
- Vertex B: [tex]\((0, 9)\)[/tex]
- Vertex C: [tex]\((0, -8)\)[/tex]
- Vertex D: [tex]\((-1, -8)\)[/tex]
2. Calculate the lengths of the sides:
- Side AB: The distance between points A and B can be determined using the distance formula [tex]\( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex].
[tex]\[
\text{Length of AB} = \sqrt{(0 - (-1))^2 + (9 - 9)^2} = \sqrt{1^2 + 0^2} = \sqrt{1} = 1.0 \quad \text{units}
\][/tex]
- Side BC: The distance between points B and C.
[tex]\[
\text{Length of BC} = \sqrt{(0 - 0)^2 + (-8 - 9)^2} = \sqrt{0 + (-17)^2} = \sqrt{289} = 17.0 \quad \text{units}
\][/tex]
- Side CD: The distance between points C and D.
[tex]\[
\text{Length of CD} = \sqrt{(-1 - 0)^2 + (-8 - (-8))^2} = \sqrt{(-1)^2 + 0^2} = \sqrt{1} = 1.0 \quad \text{units}
\][/tex]
- Side DA: The distance between points D and A.
[tex]\[
\text{Length of DA} = \sqrt{(-1 - (-1))^2 + (9 - (-8))^2} = \sqrt{0 + 17^2} = \sqrt{289} = 17.0 \quad \text{units}
\][/tex]
3. Calculate the perimeter of the rectangle:
- In a rectangle, opposite sides are equal. Therefore, the perimeter [tex]\( P \)[/tex] is given by:
[tex]\[
P = 2 \times (\text{Length of AB} + \text{Length of BC})
\][/tex]
Substituting the side lengths we calculated:
[tex]\[
P = 2 \times (1.0 + 17.0) = 2 \times 18.0 = 36.0 \quad \text{units}
\][/tex]
Therefore, the perimeter of rectangle ABCD is [tex]\( \boxed{36.0} \)[/tex] units.
