Question

Suppose a parallelogram is drawn on a coordinate plane. 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 find the perimeter of the parallelogram defined by the vertices (2,1), (2,3), (4,4), and (4,2), we will first calculate the lengths of the sides.

1. **Identify the vertices:**
Let the vertices be:
- \( A(2, 1) \)
- \( B(2, 3) \)
- \( C(4, 4) \)
- \( D(4, 2) \)

2. **Calculate the lengths of the sides:**
- **Length of side AB:**
\[
AB = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 4} = \sqrt{4} = 2
\]
- **Length of side BC:**
\[
BC = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]
- **Length of side CD:**
\[
CD = \sqrt{(4 - 4)^2 + (2 - 4)^2} = \sqrt{0 + 4} = \sqrt{4} = 2
\]
- **Length of side DA:**
\[
DA = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]

3. **Perimeter calculation:**
The perimeter \( P \) of the parallelogram can be calculated as:
\[
P = AB + BC + CD + DA = 2 + \sqrt{5} + 2 + \sqrt{5} = 4 + 2\sqrt{5}
\]

4. **Evaluate \( 2\sqrt{5} \):**
The approximate value of \( \sqrt{5} \) is about 2.236. Thus,
\[
2\sqrt{5} \approx 2 \times 2.236 = 4.472
\]

5. **Calculating the perimeter:**
Adding this to 4 gives:
\[
P \approx 4 + 4.472 = 8.472
\]

Rounding this to the nearest tenth:
\[
P \approx 8.5
\]

Thus, the perimeter of the parallelogram is \( \boxed{8.5} \).