Question

Parallelogram vertices
(2,1)
(2,3)
(4,4)
(4,2)
compute perimeter
Round to nearest tenth

Answer 1

To find the perimeter of a parallelogram, we first need to determine the lengths of its sides. The vertices of the parallelogram given are:

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

We can calculate the lengths of each side by using the distance formula between the vertices. The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

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

### Calculating Lengths of the Sides:

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

2. **Length of side BC**:
\[
BC = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]

3. **Length of side CD**:
\[
CD = \sqrt{(4 - 4)^2 + (2 - 4)^2} = \sqrt{0 + 4} = \sqrt{4} = 2
\]

4. **Length of side DA**:
\[
DA = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]

### Perimeter of the Parallelogram:

The perimeter \( P \) of a parallelogram can be calculated as:
\[
P = 2(AB + BC)
\]
Using our calculations:
\[
P = 2(2 + \sqrt{5})
\]

### Approximating \(\sqrt{5}\):
The approximate value of \(\sqrt{5}\) is about \(2.236\).

Now substituting back into the perimeter equation:
\[
P = 2(2 + 2.236) = 2(4.236) = 8.472
\]

### Rounding to the Nearest Tenth:
\[
P \approx 8.5
\]

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