The coordinates of the vertices of a polygon are A(0, 0), B(7, 0), C(9, 7), and D(2,7). Find the area and determine whether it is a parallelogram, rectangle, or square.
To find the area of the polygon and determine its shape, we can use the coordinates of the vertices to analyze its sides and angles.
1. Plot the points on a coordinate plane:
```
A(0, 0) -- B(7, 0)
| |
D(2, 7) -- C(9, 7)
```
2. Calculate the lengths of the sides:
- Side AB: Distance between points A and B = √((7 - 0)^2 + (0 - 0)^2) = √49 = 7
- Side BC: Distance between points B and C = √((9 - 7)^2 + (7 - 0)^2) = √50 ≈ 7.07
- Side CD: Distance between points C and D = √((2 - 9)^2 + (7 - 7)^2) = √49 = 7
- Side DA: Distance between points D and A = √((0 - 2)^2 + (0 - 7)^2) = √53 ≈ 7.28
3. Check if opposite sides are parallel:
- AB and CD: The slopes of AB and CD are equal, which means they are parallel.
- BC and DA: The slopes of BC and DA are equal, so they are parallel.
4. Check if adjacent sides are perpendicular:
- AB and BC: The slope of AB is 0, and the slope of BC is undefined because it is a vertical line. Therefore, AB and BC are perpendicular.
- BC and CD: The slopes of BC and CD are both 0, so they are parallel.
- CD and DA: The slope of CD is undefined, and the slope of DA is 0. Therefore, CD and DA are perpendicular.
- DA and AB: The slopes of DA and AB are both 0, so they are parallel.
Based on the above analysis, we can conclude that the polygon formed by the given coordinates is a rectangle. Additionally, since all sides of a rectangle are equal, we can also consider it a square.
5. Calculate the area of the rectangle:
Since opposite sides of a rectangle are parallel and equal, we can use the formula for the area of a rectangle: Area = length × width.
Length = AB = 7 (the distance between A(0, 0) and B(7, 0))
Width = BC = 7 (the distance between B(7, 0) and C(9, 7))
Area = 7 × 7 = 49 square units.
Therefore, the given polygon is a rectangle and its area is 49 square units.