Find a formula for the area of a parallelogram whose vertices, in order, are

Question

P, Q, R & S.

Answer 1

To find the area of a parallelogram, we need the length of the base and the height.

Let's start with the length of the base. The base of the parallelogram is the line segment PQ or SR. We can find the length of the base PQ by using the distance formula:

Length of PQ = sqrt((Qx - Px)^2 + (Qy - Py)^2)

Similarly, the length of the base SR is:

Length of SR = sqrt((Rx - Sx)^2 + (Ry - Sy)^2)

Next, let's find the height of the parallelogram. The height is the perpendicular distance between the base and the opposite side. In this case, the opposite side would be RS or PQ. We can find the distance between a point and a line using the formula for the area of a triangle (since the height is equivalent to the height of a triangle). Let's calculate the height using the distance formula for the point P and the line SR:

Height = abs((Sx - Rx) * (Py - Ry) - (Px - Rx) * (Sy - Ry)) / sqrt((Sx - Rx)^2 + (Sy - Ry)^2)

Finally, we can calculate the area of the parallelogram using the formula:

Area = base * height

Area = Length of PQ * Height

Area = Length of SR * Height

So, the formula for the area of a parallelogram given the coordinates of P, Q, R, and S is:

Area = sqrt((Qx - Px)^2 + (Qy - Py)^2) * abs((Sx - Rx) * (Py - Ry) - (Px - Rx) * (Sy - Ry)) / sqrt((Sx - Rx)^2 + (Sy - Ry)^2)