Question

Suppose a rectangle is drawn on a coordinate plane. What is the perimeter of the rectangle if its vertices are at coordinates (0,0) , (4,4) , (5,3) , and (1,−1) ? Round your answer to the nearest tenth.(1 point) The perimeter is approximately units.

Answer 1

To find the perimeter of the rectangle with vertices at (0,0), (4,4), (5,3), and (1,-1), we need to first confirm that these points actually form a rectangle and then calculate its perimeter.

1. **Check if the four points form a rectangle:**
We need to ensure opposite sides are equal in length, and the angles between the sides are right angles.

- Calculate the distances between each pair of points:

- Distance from (0,0) to (4,4):
\[
d_1 = \sqrt{(4 - 0)^2 + (4 - 0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]

- Distance from (4,4) to (5,3):
\[
d_2 = \sqrt{(5 - 4)^2 + (3 - 4)^2} = \sqrt{1 + 1} = \sqrt{2}
\]

- Distance from (5,3) to (1,-1):
\[
d_3 = \sqrt{(1 - 5)^2 + (-1 - 3)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]

- Distance from (1,-1) to (0,0):
\[
d_4 = \sqrt{(0 - 1)^2 + (0 + 1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]

Now we see that opposite sides are equal, resulting in:
- Side 1: \(d_1 = 4\sqrt{2}\)
- Side 2: \(d_2 = \sqrt{2}\)

2. **Calculate the perimeter:**
A rectangle's perimeter \(P\) can be calculated using the formula:
\[
P = 2 \times (\text{length} + \text{width})
\]

Here, we can take the lengths we calculated:
- Length = \(4\sqrt{2}\)
- Width = \(\sqrt{2}\)

The perimeter can now be calculated as:
\[
P = 2 \times (4\sqrt{2} + \sqrt{2}) = 2 \times (5\sqrt{2}) = 10\sqrt{2}
\]

Now we need to approximate \(10\sqrt{2}\) using \(\sqrt{2} \approx 1.414\):
\[
10\sqrt{2} \approx 10 \times 1.414 \approx 14.14
\]

3. **Final answer:**
Rounding to the nearest tenth, the perimeter is approximately \(14.1\) units.

Thus, the perimeter of the rectangle is approximately **14.1 units**.