To find the perimeter of rectangle ABCD with the given coordinates \( A(0, 0) \) and \( B(3, 3) \), we need to determine the coordinates of the other two vertices \( C \) and \( D \).
Since the diagonal distance from point \( B \) to point \( D \) is given as 8 units, we can use this information along with the coordinates of \( B \) to find \( D \).
Step 1: Diagonal distance
The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the distance formula:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 2: Let’s determine point \( D \)
We need to place point \( D \) such that the distance from \( B(3, 3) \) to \( D(x_D, y_D) \) equals 8. Using the distance formula:
\[ \sqrt{(x_D - 3)^2 + (y_D - 3)^2} = 8 \]
Squaring both sides gives us:
\[ (x_D - 3)^2 + (y_D - 3)^2 = 64 \]
Possible coordinates for \( D \)
To find a specific location of \( D \), we can choose to keep it on either the same x or y level as \( A \) or \( C \). For simplicity, let’s choose points such that \( D \) is in relative coordinates to \( B \) continuing vertically or horizontally.
Finding the sides of the rectangle
If we take \( D(x_D, 0) \) such that the length of the rectangle can be found easily:
Let’s find \( D \) with \( y_D = 0 \): \[ (x_D - 3)^2 + (0 - 3)^2 = 64 \] \[ (x_D - 3)^2 + 9 = 64 \] \[ (x_D - 3)^2 = 55 \] \[ x_D - 3 = \pm \sqrt{55} \] \[ x_D = 3 \pm \sqrt{55} \] We can choose \( D = (3 + \sqrt{55}, 0) \).
The rectangle’s \( A \) and \( D \) gives us one dimension, the width between \( A \) to \( D \) is:
- \( AD: |(3 + \sqrt{55} - 0, 0) - (0, 0)| \)
The width \( AD = 3 + \sqrt{55} \).
Finding the height of the rectangle AB
The height \( AB \):
- \( AB: |(3 - 0)| = 3 \)
Calculate the perimeter
The perimeter \( P \) of the rectangle can be calculated by: \[ P = 2 \times (\text{width} + \text{height}) = 2 \times ((3 + \sqrt{55}) + 3) \] Given \( \sqrt{55} \approx 7.4 \): \[ 3 + \sqrt{55} \approx 3 + 7.4 = 10.4 \] So: \[ P \approx 2 \times (10.4 + 3) = 2 \times 13.4 = 26.8 \]
Thus, the approximate perimeter of rectangle ABCD, rounded to the tenths place, is: \[ \boxed{26.8} \]
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