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

A rectangle is drawn on a coordinate plane with vertices (0,0) , (2,2) , (9,−4) , and (7,−6) . Using the coordinates, what is the approximate area of the rectangle? (1 point) Responses 26.08 units2 26.07 units squared 24.1 units2 24.1 units squared 52.2 units2 52.2 units squared 12.05 units2

Answer 2

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

1. **Determining the Distances (Sides of the Rectangle)**

We can calculate the lengths of the sides by finding the distance between each pair of adjacent vertices. Using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

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

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

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

- Between (1, -1) and (0, 0):
\[
d_4 = \sqrt{(0 - 1)^2 + (0 - (-1))^2} = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]

2. **Confirming Rectangle Properties**
In a rectangle, opposite sides are equal, and adjacent sides are perpendicular. Here, we have:
- Two sides of length \(4\sqrt{2}\)
- Two sides of length \(\sqrt{2}\)

Since the sides are not only equal in pairs but also appear to meet at right angles (inferred from the different distances), we can deduce that these points do form a rectangle.

3. **Calculating the Perimeter**
The perimeter \(P\) of a rectangle is given by:
\[
P = 2(\text{Length} + \text{Width}) = 2(4\sqrt{2} + \sqrt{2}) = 2(5\sqrt{2}) = 10\sqrt{2}
\]

4. **Numerical Calculation**
Next, let's evaluate \(10\sqrt{2}\) numerically:
\[
\sqrt{2} \approx 1.414
\]
\[
10\sqrt{2} \approx 10 \times 1.414 = 14.14
\]

5. **Final Rounding**
Rounding to the nearest tenth gives us:
\[
\text{Perimeter} \approx 14.1
\]

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