9. A polygon on a coordinate plane has vertices A(2, 10), B(2, 5), C(12, 1), and D(12, 10). Find the area of the polygon.

Answer 1

To find the area of the polygon, we can split it into two triangles by drawing a diagonal from A(2, 10) to C(12, 1).

First, let's find the length of the base of the triangle formed by points A, B, and C. This is the horizontal difference between the x-coordinates of A and C:
Base = 12 - 2 = 10 units

Next, we need to find the height of the triangle. This is the vertical difference between the y-coordinate of A and the y-coordinate of C. This can be calculated using the Pythagorean theorem by drawing a vertical line from point B(2, 5) to the line containing points A and C:
Height = sqrt((12 - 2)^2 + (1 - 10)^2)
Height = sqrt(100 + 81)
Height = sqrt(181) ≈ 13.45 units

Now, we can calculate the area of the triangle formed by points A, B, and C:
Area = 0.5 * base * height
Area = 0.5 * 10 * 13.45
Area = 0.5 * 134.5
Area = 67.25 square units

Since the polygon consists of two congruent triangles, the total area of the polygon is 67.25 * 2 = 134.5 square units.