Question

Using vertices with the coordinates (2,1), (2,3), (4,4), and (4,2), compute the perimeter. Round your answer to the nearest tenth

Answer 1

To compute the perimeter of a polygon defined by the vertices \((2,1)\), \((2,3)\), \((4,4)\), and \((4,2)\), we first need to find the lengths of the sides between each pair of consecutive vertices.

1. **Distance between \((2,1)\) and \((2,3)\)**:
\[
d_1 = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 4} = \sqrt{4} = 2
\]

2. **Distance between \((2,3)\) and \((4,4)\)**:
\[
d_2 = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236
\]

3. **Distance between \((4,4)\) and \((4,2)\)**:
\[
d_3 = \sqrt{(4 - 4)^2 + (2 - 4)^2} = \sqrt{0 + 4} = \sqrt{4} = 2
\]

4. **Distance between \((4,2)\) and \((2,1)\)**:
\[
d_4 = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236
\]

Now, we calculate the total perimeter \(P\) by adding the lengths of all sides:

\[
P = d_1 + d_2 + d_3 + d_4 = 2 + \sqrt{5} + 2 + \sqrt{5}
\]
\[
P = 4 + 2\sqrt{5} \approx 4 + 2 \times 2.236 \approx 4 + 4.472 \approx 8.472
\]

Rounding this to the nearest tenth:

\[
P \approx 8.5
\]

Thus, the perimeter of the shape formed by the given vertices is approximately **8.5**.