Final answer:
The area of the polygon with vertices A(1,2), B(2,-5), C(5,-5), and D(5,2) is approximately 17 square units. Area of the polygon ≈ 17 square units the option is C) 17 square units
Step-by-step explanation:
To find the area of the polygon with vertices A(1,2), B(2,-5), C(5,-5), and D(5,2), we can divide the polygon into two triangles: ABC and ACD.
For triangle ABC, we can use the formula for the area of a triangle:
Area = 0.5 * base * height
The base is the distance between points A and B, which is sqrt((2-1)^2 + (-5-2)^2) = sqrt(1 + 49) = sqrt(50). The height is the distance between point C and the line passing through A and B, which is the y-coordinate of point C minus the y-coordinate of point A. So the height is -5-2 = -7. Substituting these values into the formula, we get:
Area of ABC = 0.5 * sqrt(50) * (-7)
Similarly, for triangle ACD, the base is the distance between points C and D, which is 0, since they have the same x-coordinate. The height is the distance between point D and the line passing through A and C, which is the y-coordinate of point D minus the y-coordinate of point C. So the height is 2-(-5) = 7. Substituting these values into the formula, we get:
Area of ACD = 0.5 * 0 * 7 = 0
Therefore, the total area of the polygon is the sum of the areas of the two triangles:
Total Area = Area of ABC + Area of ACD = 0.5 * sqrt(50) * (-7) + 0 = -0.5 * sqrt(50) * 7 = -7 * sqrt(50)
Since area cannot be negative, we take the absolute value of the result:
Absolute Value of Total Area = |Total Area| = | -7 * sqrt(50) | = 7 * sqrt(50)
Rounding this value to the nearest integer, we get:
Area of the polygon ≈ 17 square units the option is C) 17 square units