Question

Suppose a parallelogram is drawn on a coordinate plane. Using vertices with the coordinates (0,0) , (2,3) , (5,2) , and (3,−1) , compute the perimeter. Round your answer to the nearest tenth.(1 point) The perimeter is approximately units.

Answer 1

To find the perimeter of the parallelogram formed by the given vertices, we first need to identify the sides of the parallelogram using the given points: \( A(0,0) \), \( B(2,3) \), \( C(5,2) \), and \( D(3,-1) \).

The perimeter \( P \) of a parallelogram can be computed as \( P = 2(a + b) \), where \( a \) and \( b \) are the lengths of the sides.

First, we need to find the lengths of the sides \( AB \) and \( BC \).

1. Calculate the length of \( AB \):
\[
AB = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}
\]

2. Calculate the length of \( BC \):
\[
BC = \sqrt{(5 - 2)^2 + (2 - 3)^2} = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}
\]

Both pairs of opposite sides are equal in length; so, \( CD \) will also be \( \sqrt{13} \), and \( AD \) will also be \( \sqrt{10} \).

Now, we can use the lengths of the two distinct sides to calculate the perimeter:
\[
P = 2(AB + BC) = 2(\sqrt{13} + \sqrt{10})
\]

Using approximate values:
\[
\sqrt{13} \approx 3.60555 \quad \text{and} \quad \sqrt{10} \approx 3.16228
\]
Thus,
\[
P \approx 2(3.60555 + 3.16228) = 2(6.76783) \approx 13.53566
\]

Rounding to the nearest tenth, the perimeter of the parallelogram is approximately:
\[
\boxed{13.5} \text{ units.}
\]