To find the perimeter of rectangle ABCD, we first need to identify the coordinates of points C and D, based on the given information.
Step 1: Identify Points A and B
- Point A (0, 0)
- Point B (3, 3)
Step 2: Find the Length of Side AB
The coordinates A(0,0) and B(3,3) represent a diagonal of the rectangle. The distance formula between two points (x₁, y₁) and (x₂, y₂) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Calculating the distance AB:
\[ d = \sqrt{(3 - 0)^2 + (3 - 0)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24 \]
This means the length of side AB is approximately 4.24 units.
Step 3: Calculate the Length of Side AD
Given that the diagonal distance from point B to point D is 8 units, we represent the coordinates of D as (x, 0) and the side length AD will be the height of the rectangle.
Step 4: Pythagorean Theorem to Find Side AD
The diagonal BD can be described as:
\[ BD = \sqrt{(x - 3)^2 + (0 - 3)^2} \]
We know that the distance BD is 8 units, so we have:
\[ \sqrt{(x - 3)^2 + 9} = 8 \]
Squaring both sides results in:
\[ (x - 3)^2 + 9 = 64 \] \[ (x - 3)^2 = 55 \]
Taking the square root gives us:
\[ x - 3 = \pm \sqrt{55} \Rightarrow x = 3 \pm \sqrt{55} \]
Step 5: Plugging in Values of x
We will consider \( x = 3 + \sqrt{55} \) because we're computing lengths and will be taking positive values.
To find the length AD, we have:
\[ AD = |0 - 3| = 3 \]
Step 6: Calculate the Rectangle's Perimeter
Now we have:
- Length AB (width) = 4.24 units
- Length AD (height) = √55 (since we calculated the other way):
Compute the perimeter P:
\[ P = 2 \times (width + height) = 2 \times (4.24 + \sqrt{55}) = 2 \times (4.24 + 7.42) = 2 \times 11.66 = 23.32 \]
Final Answer
The perimeter of rectangle ABCD is approximately \( 23.3 \) units. The closest answer from the options provided is:
22.1 units (By approximation, chosen the closest rational value).