Find a formula for the area of a parallelogram whose vertices, in order, are
P, Q, R & S.
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)
To find the formula for the area of a parallelogram, we need to know the lengths of its base and height.
Given the vertices of the parallelogram are P, Q, R, and S, we can find its base and height by considering the lines formed by the vertices.
Let's start by identifying the vectors formed by two pairs of vertices. We will consider the vectors formed by the sides PR and PS.
The vector PR is represented as vector P to R, and the vector PS is represented as vector P to S.
To find the vector PR, we subtract the coordinates of point P from point R:
PR = R - P
Similarly, to find the vector PS, we subtract the coordinates of point P from point S:
PS = S - P
Now, we have two vectors, PR and PS.
To find the height of the parallelogram, we need to find the component of vector PR that is perpendicular to vector PS. This can be done using the dot product of PR and PS.
The dot product of two vectors A = (A₁, A₂) and B = (B₁, B₂) is given by:
A · B = A₁ * B₁ + A₂ * B₂
In our case, A = PR = (PR₁, PR₂) and B = PS = (PS₁, PS₂).
The dot product of PR and PS is:
PR · PS = PR₁ * PS₁ + PR₂ * PS₂
Next, we calculate the magnitude or length of vector PS, which is given by:
|PS| = sqrt(PS₁² + PS₂²)
We then divide the dot product (PR · PS) by the magnitude of vector PS (|PS|) to obtain the height (h):
h = (PR · PS) / |PS|
Now, we need to find the length of the base. To do this, we need the magnitude or length of vector PR, which is given by:
|PR| = sqrt(PR₁² + PR₂²)
The length of the base of the parallelogram is equal to the magnitude of vector PR.
Finally, the formula for the area (A) of a parallelogram is given by:
A = base * height = |PR| * h
Substituting the expressions for base and height:
A = |PR| * [(PR · PS) / |PS|]
This is the formula for the area of a parallelogram in terms of its vertices P, Q, R, and S.