Practice Parallelograms
A parallelogram is a quadrilateral whose opposite sides are parallel and which has two pairs of congruent adjacent sides. The opposite sides are called the bases of the parallelogram. The line that connects the midpoints of the pairs of opposite sides, called a the diagonal, is also a base of the parallelogram. The area and perimeter of a rectangular parallelogram can be found using formulas similar to those used for triangles. a The determinant of a matrix is a number that can be computed from the elements of the matrix. It is used in various calculations and is useful to know when solving systems of linear equations, among other things. Determinants are used in many areas of mathematics. They are used to find the volume of a parallelepiped, solve systems of linear equations and find eigenvalues. Determinants can also be used to find the area under a curve and find the centroid (center of mass) of a triangle or quadrilateral by taking the average of all its vertices. Let’s begin working on this problem. The first thing we need to be careful about is knowing that we want to take a determinant, but we need to be careful. Determinants of pairs of vectors make sense, but those of points do not. In this example, we have four points that form the basis for a parallelogram. To calculate the area of the parallelogram, we need to compute vectors from these points. So we have taken the vectors that connect the endpoints of the parallelogram. You will see that vector 6, 1 is coming from point 1, 1 in the original parallelogram and 7, 2. So, vector 6. 1 is equivalent to the difference between point 7, 2 1, and point 1, 1. The same way, vector 5, 2 is equivalent to the difference of the original point 6, 3 and the base point 1. 1. Now that we have these two vectors, the area of our parallelogram is going to be equal to the determinant of the two vectors. We should be careful because the area will be determined by the value of the determinant (positive or negative). It is time to compute this determinant. Now we can find six times two minus five. This gives us twelve minus five, which is seven. IS It is a positive one. This plus or minus, we take to be positive. If we had computed our determinant by transposing the rows, then we might have found a negative 7. To area be positive, we would choose 7. The tricky part of this problem is that the original endpoints of our parallelogram are not what is important for figuring out its area. What we need to find are the vectors connecting those original endpoints together. 1 We computed the values of 6, 1 and 5, 2 and then taking their determinant resulted in the area of the parallelogram. That’s it. [Graph][Graph]
