To find the coordinates of points C and D, we need to first determine the lengths of the sides of rectangle ABCD. Given the coordinates of points A and B, let's analyze the positioning.
Point A is at (0, 0), and point B is at (3, 3). The distance from point B to point D is given as 8 units. Since a rectangle's opposite sides are equal and parallel, we can determine points C and D by understanding their relations to points A and B.
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Finding the coordinates of D: Let's denote:
- Point A: \( A(0,0) \)
- Point B: \( B(3,3) \)
Assuming the sides of the rectangle are aligned with the axes, we can deduce that:
- Point C would be vertically aligned with B, possibly at \( C(3, y_C) \).
- Point D would be horizontally aligned with A, possibly at \( D(x_D, 0) \).
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Calculating the length of CD (vertical side of the rectangle): Since we have points B and D, we use the distance formula: \[ \text{Distance}(B,D) = \sqrt{(x_D - 3)^2 + (0 - 3)^2} \] Given that this distance is 8: \[ 8 = \sqrt{(x_D - 3)^2 + 9} \] Squaring both sides gives: \[ 64 = (x_D - 3)^2 + 9 \] \[ 64 - 9 = (x_D - 3)^2 \] \[ 55 = (x_D - 3)^2 \] Taking the square root: \[ x_D - 3 = \pm \sqrt{55} \] So: \[ x_D = 3 \pm \sqrt{55} \]
This gives us two possible coordinates for point D, \( D(3 + \sqrt{55}, 0) \) or \( D(3 - \sqrt{55}, 0) \).
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Calculating the dimensions of the rectangle: To find the width of the rectangle (which is the distance between A and D), we can use \( |x_D| \): \[ |x_D| = 3 + \sqrt{55} \quad (\text{choosing the positive root for a valid rectangle context}) \] The width (CD) = \( |y_C| = 3 \) (as the y-coordinate must be the same for points B and C).
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Calculating the perimeter (P) of the rectangle: The formula for the perimeter \( P \) of a rectangle is: \[ P = 2(\text{length} + \text{width}) \] Here, length = \( 8 \) and width = \( 3 \): \[ P = 2(8 + 3) = 2 \times 11 = 22 \]
The approximate perimeter of the rectangle ABCD is 22.0 (to the tenths place).
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