Question

Find the area of polygon WXYZ with the vertices W(-3,-7), X(-5,5), Y(3,2), and Z(6,-6).

Answer 1

To find the area of the polygon WXYZ, divide it into two triangles: WXY and XYZ. Find the lengths of the triangle sides using the distance formula, and then use the formula for the area of a triangle.

Step-by-step explanation:

To find the area of polygon WXYZ, we can use the formula for the area of a quadrilateral. We will divide the quadrilateral into two triangles and find the area of each triangle, then add them together.

Triangle WXY:

1. Find the length of WX using the distance formula: sqrt((-5 - (-3))^2 + (5 - (-7))^2) = sqrt(2^2 + 12^2) = sqrt(4 + 144) = sqrt(148) = 2sqrt(37)
2. Find the length of XY using the distance formula: sqrt((3 - (-5))^2 + (2 - 5)^2) = sqrt(8^2 + (-3)^2) = sqrt(64 + 9) = sqrt(73)
3. Calculate the area of triangle WXY using the formula: (1/2) * base * height = (1/2) * 2sqrt(37) * sqrt(73) = sqrt(37 * 73) = sqrt(2701)

Triangle XYZ:

1. Find the length of XY using the distance formula: sqrt((3 - (-5))^2 + (2 - 5)^2) = sqrt(8^2 + (-3)^2) = sqrt(64 + 9) = sqrt(73)
2. Find the length of XZ using the distance formula: sqrt((3 - 6)^2 + (2 - (-6))^2) = sqrt((-3)^2 + 8^2) = sqrt(9 + 64) = sqrt(73)
3. Calculate the area of triangle XYZ using the formula: (1/2) * base * height = (1/2) * sqrt(73) * sqrt(73) = (1/2) * 73 = 36.5

Finally, add the areas of the two triangles together: sqrt(2701) + 36.5 = sqrt(2701) + 36.5 = approximately 57.2 square units.