What is the area of a parallelogram with corners at these coordinates on a plane: (-2,-1), (-12,-4), (9,-4), (-1,-7)?

Answer 1

Let see the figure which shows the parallelogram with given vertices

As the diagonal of a parallelogram BC divides into two congruent triangles, which implies the triangles have same area.

So, the area of the parallelogram will be 2 times the area of the triangle ABC.

We know from analytical geometry that the area of a triangle with given vertices is

where #(x_A,y_A)# ,#(x_B,y_B)#,#(x_C,y_C)# are the coordinates of points A,B,C.

Now we just have to replace the values in the formula above and find the value for #E_(ABC)#

Hence the area of the parallelogram is

#E_(ABCD)=2*E_(ABC)#

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Answer 2

To find the area of a parallelogram given its vertices, you can use the determinant of a matrix formed by the coordinates of its sides. Given the vertices ((-2, -1)), ((-12, -4)), ( (9, -4)), and ((-1, -7)), you can calculate the area by taking the vector that represents one side and the vector that represents an adjacent side, and then calculate the determinant of these vectors.

First, let's organize the points into vectors. Choose two pairs of adjacent sides. For simplicity, let's use ((-2, -1)) to ((-12, -4)) as one vector and ((-2, -1)) to ((-1, -7)) as the other vector.

Calculate the vectors:

  • Vector A (((-2, -1)) to ((-12, -4))): ((-12 + 2, -4 + 1) = (-10, -3))
  • Vector B (((-2, -1)) to ((-1, -7))): ((-1 + 2, -7 + 1) = (1, -6))

The area of the parallelogram is the absolute value of the determinant of the matrix formed by these vectors:

[ \text{Area} = \left| \begin{array}{cc} -10 & 1 \ -3 & -6 \ \end{array} \right| = \left| ( -10 \times -6 ) - ( 1 \times -3 ) \right| = \left| 60 + 3 \right| = \left| 63 \right| = 63 ]

So, the area of the parallelogram is 63 square units.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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