Question

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

Answer 1

To find the perimeter of the rectangle defined by the vertices at coordinates (0,0), (4,4), (5,3), and (1,-1), we first verify if the given points form a rectangle.

1. **Calculate the distances between all pairs of points** to check if the distances correspond to a rectangle's properties (two sets of equal lengths).

- Distance between (0,0) and (4,4):
\[
d = \sqrt{(4-0)^2 + (4-0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]

- Distance between (0,0) and (5,3):
\[
d = \sqrt{(5-0)^2 + (3-0)^2} = \sqrt{25 + 9} = \sqrt{34}
\]

- Distance between (0,0) and (1,-1):
\[
d = \sqrt{(1-0)^2 + (-1-0)^2} = \sqrt{1 + 1} = \sqrt{2}
\]

- Distance between (4,4) and (5,3):
\[
d = \sqrt{(5-4)^2 + (3-4)^2} = \sqrt{1 + 1} = \sqrt{2}
\]

- Distance between (4,4) and (1,-1):
\[
d = \sqrt{(1-4)^2 + (-1-4)^2} = \sqrt{9 + 25} = \sqrt{34}
\]

- Distance between (5,3) and (1,-1):
\[
d = \sqrt{(1-5)^2 + (-1-3)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]

2. **Summarizing the distances**:
- \(d((0,0),(4,4)) = 4\sqrt{2}\)
- \(d((0,0),(5,3)) = \sqrt{34}\)
- \(d((0,0),(1,-1)) = \sqrt{2}\)
- \(d((4,4),(5,3)) = \sqrt{2}\)
- \(d((4,4),(1,-1)) = \sqrt{34}\)
- \(d((5,3),(1,-1)) = 4\sqrt{2}\)

From the computed distances, we find pairs of equal lengths:
- \(4\sqrt{2}\) appears twice.
- \(\sqrt{34}\) appears twice.
- \(\sqrt{2}\) appears twice.

Thus, the opposite sides are equal in length.

3. **Calculate the perimeter**:
The perimeter \(P\) of a rectangle is given by:
\[
P = 2 \times (\text{length} + \text{width})
\]
Here, we can take the lengths as \(4\sqrt{2}\) and \(\sqrt{34}\).
Thus:
\[
P = 2 \times (4\sqrt{2} + \sqrt{34})
\]

4. **Calculating** the numerical value:
- First calculate \(4\sqrt{2}\):
\[
\sqrt{2} \approx 1.414 \Rightarrow 4\sqrt{2} \approx 4 \times 1.414 \approx 5.656
\]

- Next calculate \(\sqrt{34}\):
\[
\sqrt{34} \approx 5.831
\]

Putting these together:
\[
4\sqrt{2} + \sqrt{34} \approx 5.656 + 5.831 \approx 11.487
\]

Thus, the perimeter \(P\) would be:
\[
P \approx 2 \times 11.487 \approx 22.974
\]

5. **Rounding** the answer to the nearest tenth:
\[
P \approx 23.0
\]

The final answer for the perimeter of the rectangle is **23.0**.