8. Determine the most precise name for the quadrilateral & find the area. (2 points)
coordinates: A(3,4), B(8,1), C(2,-9), D(-3,-6)
The most precise name for the quadrilateral would be a trapezoid, since it has one pair of parallel sides (AB and CD).
To find the area of the trapezoid, we can split it into two triangles: triangle ABC and triangle ACD.
First, we find the lengths of the bases of the trapezoid by calculating the distance between the points:
AB = sqrt((8-3)^2 + (1-4)^2) = sqrt(25 + 9) = sqrt(34)
CD = sqrt((2+3)^2 + (-9+6)^2) = sqrt(25 + 9) = sqrt(34)
Next, we find the height of the trapezoid by calculating the distance between point B and line CD. The formula for the distance between a point (x1, y1) and a line Ax + By + C =0 is given by:
h = |Ax1 + By1 + C| / sqrt(A^2 + B^2)
In this case, the equation for line CD is 5x + 3y + 27 = 0. Plugging in point B(8,1), we get:
h = |5(8) + 3(1) + 27| / sqrt(5^2 + 3^2)
h = |40 + 3 + 27| / sqrt(34)
h = |70| / sqrt(34)
h = 70 / sqrt(34)
Now, we can calculate the area of the trapezoid by adding the areas of the two triangles:
Area of trapezoid = Area of triangle ABC + Area of triangle ACD
Area of trapezoid = (1/2 * AB * h) + (1/2 * CD * h)
Area of trapezoid = (1/2 * sqrt(34) * 70 / sqrt(34)) + (1/2 * sqrt(34) * 70 / sqrt(34))
Area of trapezoid = 70
Therefore, the area of the trapezoid is 70 square units.