Question

A(5, −3)​ , cap b times open paren 10 comma negative 3 close paren$B\left(10,\ -3\right)$B(10, −3)​ , cap c times open paren 12 comma negative 11 close paren$C\left(12,\ -11\right)$C(12, −11)​ , and cap d times open paren 7 comma negative 11 close paren$D\left(7,\ -11\right)$D(7, −11)​ .

The area of parallelogram cap A cap b cap c cap d$\parallelogram ABCD$▱ABCD​ is square units.

Answer 1

The area of parallelogram ABCD is 40 square units.

To find the area of the parallelogram formed by the points A(5,−3), B(10,−3), C(12,−11), and D(7,−11), you can use the formula that finds the area of a parallelogram formed by vectors.

Let's denote the vectors AB and AD to find the area of the parallelogram.

The formula for the area of a parallelogram formed by two vectors u and v is the magnitude of their cross product, denoted as ∣ u × v∣.

First, find the vectors AB and AD

AB =⟨x2 −x1​ ,y2 −y1)

AB =⟨10−5,−3−(−3)⟩

AB =⟨5,0⟩

AD =⟨x4 −x1 ,y4 −y1)

AD =⟨7−5,−11−(−3)⟩

AD =⟨2,−8⟩

Now, find the cross product of these vectors:

AB × AD =(5⋅−8−0⋅2)

AB × AD =(−40)

The magnitude of this cross product represents the area of the parallelogram:

Area = ∣ AB × AD ∣=∣−40∣=40 square units

Therefore, the area of parallelogram ABCD is 40 square units.