Question

Use the coordinates to compute the perimeter of a parallelogram whose vertices are located at the ordered pairs (0,0)
, (1,2)
, (2,1)
and (1,−1)
.(1 point)
Responses

7.2 units
7.2 units

8.9 units
8.9 units

2.2 units
2.2 units

4 units

Answer 1

To calculate the perimeter of the parallelogram formed by the vertices \((0,0)\), \((1,2)\), \((2,1)\), and \((1,-1)\), we first need to find the lengths of the sides.

**Step 1: Calculate the lengths of the sides.**

We can calculate the lengths of the sides using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

**Length of side 1: Between (0,0) and (1,2)**

\[
d_1 = \sqrt{(1 - 0)^2 + (2 - 0)^2} = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}
\]

**Length of side 2: Between (1,2) and (2,1)**

\[
d_2 = \sqrt{(2 - 1)^2 + (1 - 2)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]

The opposite sides of the parallelogram will have the same lengths, hence:

- The length of the side between (0,0) and (1,2) is the same as the length of the side between (1,-1) and (2,1).
- The length of the side between (1,2) and (2,1) is the same as the length of the side between (0,0) and (1,-1).

**Step 2: Calculate the perimeter.**

The perimeter \(P\) of a parallelogram is given by:

\[
P = 2 \times (d_1 + d_2)
\]

Substitute in the values we found:

\[
P = 2 \times (\sqrt{5} + \sqrt{2})
\]

Now we can approximate the numerical values:

- \(\sqrt{5} \approx 2.236\)
- \(\sqrt{2} \approx 1.414\)

So:

\[
P \approx 2 \times (2.236 + 1.414) \approx 2 \times 3.650 \approx 7.3 \text{ (approximately)}
\]

**Final Answer:**

From the options given, none match this exactly. However, rounding might lead closely to:

7.2 units

Therefore, while exact calculation suggests around \(7.3\), the closest matching option is:

**7.2 units**