Chapter 1.8, Problem 19E

To determine

To calculate: The area of the rectangle with vertices $A\left(1,3\right),B\left(5,3\right),C\left(1,-3\right)\text{ and }D\left(5,-3\right)$ and also plot the vertices on the coordinate plane.

Answer to Problem 19E

The area of the rectangle is $24\text{ sq units}$ . The rectangle so formed is provided below,

  

Explanation of Solution

Given information:

The vertices of rectangle $A\left(1,3\right),B\left(5,3\right),C\left(1,-3\right)\text{ and }D\left(5,-3\right)$ .

Formula used:

Distance $d\left(A,B\right)$ between two points $A=\left(x_1,y_1\right)\text{ and }B=\left(x_2,y_2\right)$ in the Cartesian plane is denoted by $d\left(A,B\right)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$ .

Area of rectangle is $l\times b$ where l is the length and b is the breadth of the rectangle.

Calculation:

Consider the vertices of rectangle $A\left(1,3\right),B\left(5,3\right),C\left(1,-3\right)\text{ and }D\left(5,-3\right)$ .

Toplot the points $A\left(1,3\right),B\left(5,3\right),C\left(1,-3\right)\text{ and }D\left(5,-3\right)$ in the coordinate plane.

Construct a Cartesian plane with vertical axis as y-axis and horizontal axis as x-axis.

Marks the numbers on both the axes.

Plot the points $A\left(1,3\right),B\left(5,3\right),C\left(1,-3\right)\text{ and }D\left(5,-3\right)$ .

And join them to form a rectangle.

  

Recall that area of rectangle is $l\times b$ where l is the length and b is the breadth of the rectangle.

Evaluate the length and breadth of the rectangle obtained above.

Recall that the distance $d\left(A,B\right)$ between two points $A=\left(x_1,y_1\right)\text{ and }B=\left(x_2,y_2\right)$ in the Cartesian plane is denoted by $d\left(A,B\right)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$ .

Evaluate the distance between $A\left(1,3\right)$ and $B\left(5,3\right)$ .

  $\begin{array}{c}d\left(A,B\right)=\sqrt{{\left(5-1\right)}^2+{\left(3-3\right)}^2}\\ =\sqrt{16}\\ =4\end{array}$

Evaluate the distance between $A\left(1,3\right)$ and $C\left(1,-3\right)$ .

  $\begin{array}{c}d\left(A,C\right)=\sqrt{{\left(1-1\right)}^2+{\left(-3-3\right)}^2}\\ =\sqrt{36}\\ =6\end{array}$

Therefore, length is 6 and breadth is 4 units.

Now, area of rectangle is,

  $\begin{array}{c}l\times b=6\times 4\\ =24\end{array}$

Thus, the area of the rectangle is $24\text{ sq units}$ .