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

Answer 1

To find the perimeter of the rectangle defined by the given vertices, we first need to ascertain which of these points form the rectangle, as the coordinates given do not form a rectangle by inspection.

### Step 1: Identify the vertices
The given vertices are:
- A: (0, 0)
- B: (4, 4)
- C: (5, 3)
- D: (1, -1)

We need to check if these points can indeed represent the corners of a rectangle. The diagonals of a rectangle should be equal, and adjacent sides must be perpendicular.

### Step 2: Check sides for length
We can calculate the lengths of the sides between each pair of points:

1. Distance between A (0, 0) and B (4, 4):
\[
\text{Distance} = \sqrt{(4-0)^2 + (4-0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]

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

3. Distance between A (0, 0) and D (1, -1):
\[
\text{Distance} = \sqrt{(1-0)^2 + (-1-0)^2} = \sqrt{1 + 1} = \sqrt{2}
\]

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

5. Distance between B (4, 4) and D (1, -1):
\[
\text{Distance} = \sqrt{(1-4)^2 + (-1-4)^2} = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}
\]

6. Distance between C (5, 3) and D (1, -1):
\[
\text{Distance} = \sqrt{(1-5)^2 + (-1-3)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = 4\sqrt{2}
\]

### Step 3: Find the shape
The calculated distances:
- From A to B: \( 4\sqrt{2} \)
- From A to C: \( \sqrt{34} \)
- From A to D: \( \sqrt{2} \)
- From B to C: \( \sqrt{2} \)
- From B to D: \( \sqrt{34} \)
- From C to D: \( 4\sqrt{2} \)

**Identifying pairs:**
- Sides: \( \sqrt{34} \) appears between opposite pairs of points (A to C and B to D).

The computed distances do not match a rectangle's properties (opposite sides being equal).

### Final step: Conclusion
Since the point arrangement does not create a rectangular structure, **we conclude to verify a valid rectangle.**

### Correcting the conclusion:
From the initial lineup of vertices, we do not have a rectangle, thus finding the perimeter of an assumed rectangle isn't achievable with the current set; we should check which other point configurations might yield valid rectangles or reconsider the problem statement.

**However, if the problem were stated correctly with unique rectangular properties, input point coordinates would need revision for perimeter calculation to hold true.**

Thus, the final response regarding the rectangle perimeter is undefined, as the set provided doesn't yield a rectangle dimension or structure physically alignable within defined geometric properties.