Question

In Exercises 50–53 the coordinates of points P, Q, R, and S are given. (a) Determine whether quadrilateral PQRS is a parallelogram. (b) Find the area of quadrilateral PQRS.

P(−1, 3), Q(2, 5), R(4, 1), S(1, −1).

Short answer

Part (a) The quadrilateral PQRS is a parallelogram.

Part (b) The area of quadrilateral PQRS is 16 square units.

Step-by-step solution

Part (a) Step 1. Given Information

A quadrilateral is a closed shape and a four-sided polygon that has four edges, four vertices, and four angles.

A parallelogram is defined as a quadrilateral with both pairs of opposite sides parallel and equal.

Part (a) Step 2. Determine whether quadrilateral PQRS is a parallelogram

To find that quadrilateral PQRS is a parallelogram. We have to find the vectors of opposite sides because a quadrilateral is said to be a parallelogram if the opposite sides of a quadrilateral are equal and parallel.

So,

\(\underset{PQ}{\rightarrow}=\left< 2-(-1),5-3 \right>\)

\(\underset{PQ}{\rightarrow}=\left<3,2 \right>\)

And

\(\underset{QR}{\rightarrow}=\left< 4-2,1`-5 \right>\)

\(\underset{QR}{\rightarrow}=\left<2,-4 \right>\)

And

\(\underset{RS}{\rightarrow}=\left< 1-4,(-1)-1 \right>\)

\(\underset{RS}{\rightarrow}=\left<-3,-2 \right>\)

And

\(\underset{SP}{\rightarrow}=\left< (-1)-1,3-(-1) \right>\)

\(\underset{SP}{\rightarrow}=\left<-2,4 \right>\)

The opposite vectors of a quadrilateral are scalar multiples of each other which means opposite sides of a quadrilateral are parallel.

Now, let's find the magnitude of the vectors

\(\underset{PQ}{\rightarrow}=\left<3,2 \right>\)

\(\|\overrightarrow{PQ}\|=\sqrt{2^2+3^2}\)

\(\|\overrightarrow{PQ}\|=\sqrt{4+9}\)

\(\|\overrightarrow{PQ}\|=\sqrt{13} \quad\)

And

\(\underset{QR}{\rightarrow}=\left<2,-4 \right>\)

\(\|\overrightarrow{QR}\|=\sqrt{2^2+(-4)^2}\)

\(\|\overrightarrow{QR}\|=\sqrt{4+16}\)

\(\|\overrightarrow{QR}\|=\sqrt{20} \quad\)

And

\(\underset{RS}{\rightarrow}=\left<-3,-2 \right>\)

\(\|\overrightarrow{RS}\|=\sqrt{(-3)^2+(-2)^2}\)

\(\|\overrightarrow{RS}\|=\sqrt{9+4}\)

\(\|\overrightarrow{RS}\|=\sqrt{13} \quad\)

And

\(\underset{SP}{\rightarrow}=\left<-2,4 \right>\)

\(\|\overrightarrow{SP}\|=\sqrt{(-2)^2+(4)^2}\)

\(\|\overrightarrow{SP}\|=\sqrt{4+16}\)

\(\|\overrightarrow{SP}\|=\sqrt{20} \quad\)

Thus, the opposite sides of quadrilateral PQRS are equal in length. Hence the quadrilateral PQRS is a parallelogram because opposite sides are equal and parallel.

Part (b) Step 1. Find the area

To find the area of quadrilateral PQRS, we will have to find the area of the parallelogram.

\(Area=\|\overrightarrow{PQ} \times \overrightarrow{Q R}\|\)

\(Area=\|\langle 3,2\rangle \times\langle 2,-4\rangle\|\)

\(Area=\|\langle 0,0,16\rangle\|\)

\(Area=\sqrt{0+0+16^2}\)

\(Area=16\)

Hence, the area of quadrilateral PQRS is 16 square units.