Question

Suppose a parallelogram is drawn on a coordinate plane. Using vertices with the coordinates (2,1)%0D%0A(%0D%0A2%0D%0A,%0D%0A1%0D%0A)%0D%0A , (2,3)%0D%0A(%0D%0A2%0D%0A,%0D%0A3%0D%0A)%0D%0A , (4,4)%0D%0A(%0D%0A4%0D%0A,%0D%0A4%0D%0A)%0D%0A , and (4,2)%0D%0A(%0D%0A4%0D%0A,%0D%0A2%0D%0A)%0D%0A , compute the perimeter. Round your answer to the nearest tenth.(1 point)%0D%0AThe perimeter is approximately units.

Answer 1

To compute the perimeter of the parallelogram with the given vertices, we need to find the lengths of its sides. The vertices are:

- \( A(2, 1) \)
- \( B(2, 3) \)
- \( C(4, 4) \)
- \( D(4, 2) \)

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

1. **Length of \( AB \)**:
\[
AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 4} = 2
\]

2. **Length of \( BC \)**:
\[
BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2^2) + (1^2)} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236
\]

3. **Length of \( CD \)** (which is the same as \( AB \)):
\[
CD = \sqrt{(x_D - x_C)^2 + (y_D - y_C)^2} = \sqrt{(4 - 4)^2 + (2 - 4)^2} = \sqrt{0 + 4} = 2
\]

4. **Length of \( DA \)** (which is the same as \( BC \)):
\[
DA = \sqrt{(x_A - x_D)^2 + (y_A - y_D)^2} = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236
\]

### Step 2: Compute the perimeter.

The perimeter \( P \) is given by the sum of the lengths of all sides:
\[
P = AB + BC + CD + DA = 2 + \sqrt{5} + 2 + \sqrt{5} = 4 + 2\sqrt{5}
\]
Substituting the approximate value of \( \sqrt{5} \approx 2.236 \):
\[
P \approx 4 + 2(2.236) = 4 + 4.472 = 8.472
\]

### Step 3: Round to the nearest tenth.

Rounding \( 8.472 \) to the nearest tenth gives approximately \( 8.5 \).

Thus, the perimeter of the parallelogram is approximately **8.5 units**.