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 calculate the perimeter of the rectangle given the vertices at the coordinates (0,0), (4,4), (5,3), and (1,−1), we first need to verify if these points form a rectangular shape.

**Step 1: Verify that the figure is a rectangle.**

A rectangle has opposite sides equal and all angles equal to 90 degrees. We can check this by calculating the lengths of the sides and ensuring that the diagonals are equal.

**Step 2: Calculate the lengths of the sides formed by the vertices.**

Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \):

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

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

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

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

From these calculations, we see that the lengths of opposite sides are equal:
- Two sides of length \( 4\sqrt{2} \)
- Two sides of length \( \sqrt{2} \)

**Step 3: Calculate the perimeter of the rectangle.**

The perimeter \( P \) of a rectangle is given by:
\[
P = 2(\text{length} + \text{width})
\]

Substituting the side lengths:
- Length \( = 4\sqrt{2} \)
- Width \( = \sqrt{2} \)

Calculating the perimeter:
\[
P = 2(4\sqrt{2} + \sqrt{2}) = 2(5\sqrt{2}) = 10\sqrt{2}
\]

**Step 4: Approximate the value.**

Using \( \sqrt{2} \approx 1.414 \):
\[
P \approx 10 \cdot 1.414 = 14.14
\]

**Final Answer:**
Rounding to the nearest tenth, the perimeter of the rectangle is approximately \( \boxed{14.1} \).