Q. 53 - Calculus

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(6, 3), S(4, −2)

Step-by-Step Solution

Answer

Part (a) The quadrilateral PQRS is not a parallelogram.

Part (b) The area of quadrilateral PQRS is \(\frac{49}{2}\) square units. 

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

\(\underset{QR}{\rightarrow}=\left< 6-2,3-5 \right>\)

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

And

\(\underset{RS}{\rightarrow}=\left< 4-6,(-2)-3 \right>\)

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

And

\(\underset{SP}{\rightarrow}=\left< (-1)-4,3-(-2) \right>\)

\(\underset{SP}{\rightarrow}=\left<-5,5 \right>\)

Thus, opposite vectors of a quadrilateral are not scalar multiples of each other, and opposite sides of a quadrilateral are not parallel.

Hence, the quadrilateral PQRS is not a parallelogram.  

Part (b) Step 1. Find the area

To find the area of quadrilateral PQRS, we will have to find the area of two triangles and the sum of those areas.

The formula of the area of the triangle by three points M, N, and O are given by: \(Area=\frac{1}{2}\left\|\underset{MN}{\rightarrow}\times \underset{MO}{\rightarrow} \right\|\).

So, the area of the triangle PQR is:

\(Area=\frac{1}{2}\left\|\underset{PQ}{\rightarrow}\times \underset{QR}{\rightarrow} \right\|\)

\(Area=\frac{1}{2} \left\|\left< 3,2\right>\times \left<4,-2 \right> \right\|\)

\(Area=\frac{1}{2} \left\| 0,0,14\right\|\)

\(Area=\frac{1}{2}\sqrt{0+0+(14)^{2}}\) 

\(Area=\frac{14}{2}\)

\(Area=7\)

And the area of the triangle RSP is:

\(Area=\frac{1}{2}\left\|\underset{RS}{\rightarrow}\times \underset{SP}{\rightarrow} \right\|\)

\(Area=\frac{1}{2} \left\|\left< -2,-5\right>\times \left<-5,5 \right> \right\|\)

\(Area=\frac{1}{2} \left\| 0,0,-35\right\|\)

\(Area=\frac{1}{2}\sqrt{0+0+(-35)^{2}}\) 

\(Area=\frac{35}{2}\) 

Part (b) Step 2. Find the area

The area of quadrilateral PQRS is

\(Area=\frac{35}{2}+ 7\)

\(Area=\frac{49}{2}\)

Thus, the area of quadrilateral PQRS is \(\frac{49}{2}\) square units.