Question

Rectangle ABCD has coordinates A(0,0) and B(3,3) and the diagonal distance from point B to point de is eight units using the coordinates given, and the diagonal distance what is the approximate perimeter of the rectangle round each calculation to the tenths place

Answer 1

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.

1. **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 \( D(x_D, 0) \).

2. **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 \( D(3 - \sqrt{55}, 0) \).

3. **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).

4. **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).