Question

The vertices of a quadrilateral in the coordinate plane are known. How can the perimeter of the figure be found?

Use the distance formula to find the length of each side, and then add the lengths.
Use the slope formula to find the slope of each of side, and then determine if the opposite sides are parallel.
Use the slope formula to find the slope of each of side, and then determine if the consecutive sides are perpendicular.
Use the distance formula to find the length of the sides, and then multiply two of the side lengths.

Answer 1

To find the perimeter of a quadrilateral given its vertices in the coordinate plane, you should **use the distance formula to find the length of each side, and then add the lengths**.

Here's a brief outline of the steps involved:

1. **Identify the vertices** of the quadrilateral, which we can denote as \( A(x_1, y_1) \), \( B(x_2, y_2) \), \( C(x_3, y_3) \), and \( D(x_4, y_4) \).

2. **Calculate the length of each side** using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
- For side \( AB \): \( d_{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
- For side \( BC \): \( d_{BC} = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \)
- For side \( CD \): \( d_{CD} = \sqrt{(x_4 - x_3)^2 + (y_4 - y_3)^2} \)
- For side \( DA \): \( d_{DA} = \sqrt{(x_1 - x_4)^2 + (y_1 - y_4)^2} \)

3. **Add the lengths of all sides** to find the perimeter \( P \):
\[
P = d_{AB} + d_{BC} + d_{CD} + d_{DA}
\]

Thus, the perimeter can be found by calculating the distances of each side and summing them up.