Chapter 1.8, Problem 47E

(a)

To determine

To graph: The parallelogram with the vertices $A\left(-2,-1\right),B\left(4,2\right),C\left(7,7\right)\text{ and }D\left(1,4\right)$ .

Explanation of Solution

Given information:

The vertices of the parallelogram $A\left(-2,-1\right),B\left(4,2\right),C\left(7,7\right)\text{ and }D\left(1,4\right)$ .

Graph:

The parallelogram with vertices $A\left(-2,-1\right),B\left(4,2\right),C\left(7,7\right)\text{ and }D\left(1,4\right)$ can be sketched by plotting the given points on a coordinate plane as,

  

Interpretation:

A parallelogram is a convex polygon with two pairs of parallel lines, i.e. its opposite sides are parallel and also they are of equal lengths.

A parallelogram is a quadrilateral whose diagonals bisect each other at the same point.

Opposite angles of a parallelogram are equal i.e. $\angle A=\angle C\text{ and }\angle B=\angle D$ .

The given vertices of the parallelogram satisfy all the conditions in the plotted figure.

(b)

To determine

To calculate: The mid-points of the diagonal of the parallelogram ABCD.

Answer to Problem 47E

The coordinates of mid-point of parallelogram ABCD is $\left(\frac{5}{2},3\right)$ .

Explanation of Solution

Given information:

The vertices of the parallelogram $A\left(-2,-1\right),B\left(4,2\right),C\left(7,7\right)\text{ and }D\left(1,4\right)$ .

Formula used:

Mid-point formula between two points $\text{X}\left(a_1,b_1\right)$ and $\text{Y}\left(a_2,b_2\right)$ is mathematically expressed as,

  $\left(\frac{a_1+a_2}{2},\frac{b_1+b_2}{2}\right)$

Calculation:

Consider the provided vertices of the parallelogram $A\left(-2,-1\right),B\left(4,2\right),C\left(7,7\right)\text{ and }D\left(1,4\right)$ .

By plotting the given points on the coordinate plane, we get the following figure,

  

Recall that the mid-point formula between two points $\text{X}\left(a_1,b_1\right)$ and $\text{Y}\left(a_2,b_2\right)$ is mathematically expressed as,

  $\left(\frac{a_1+a_2}{2},\frac{b_1+b_2}{2}\right)$

So, midpoint of AC is calculated as,

  $\begin{array}{c}\left(\frac{-2+7}{2},\frac{-1+7}{2}\right)=\left(\frac{5}{2},\frac{6}{2}\right)\\ =\left(\frac{5}{2},3\right)\end{array}$

Now, midpoint of BD is calculated as,

  $\begin{array}{c}\left(\frac{1+4}{2},\frac{4+2}{2}\right)=\left(\frac{5}{2},\frac{6}{2}\right)\\ =\left(\frac{5}{2},3\right)\end{array}$

Thus, the coordinates of mid-point of parallelogram ABCD is $\left(\frac{5}{2},3\right)$ .

(c)

To determine

To verify: The diagonals of the parallelogram bisects each other.

Explanation of Solution

Given information:

The vertices of the parallelogram $A\left(-2,-1\right),B\left(4,2\right),C\left(7,7\right)\text{ and }D\left(1,4\right)$ .

Formula used:

If two diagonals are intersected by each other at the same point then the diagonals bisect each other at that point.

Proof:

Consider the given vertices of the parallelogram ABCD $A\left(-2,-1\right),B\left(4,2\right),C\left(7,7\right)\text{ and }D\left(1,4\right)$ .

From part (b), it is observed that the midpoint of the diagonals of the parallelogram coincide each other.

Recall if two diagonals are intersected by each other at the same point then the diagonals bisect each other at that point.

Thus, it is proved that the diagonals of the parallelogram ABCD bisects each other at the point $\left(\frac{5}{2},3\right)$ .