Question

The coordinates of the vertices of a rectangle are \[\left( { - 3,5} \right),\left( {0, - 4} \right),\left( {3,7} \right),\left( {6, - 2} \right)\]? How do you find the area of this figure?

Answer 1

Hint:
In the given question, we have been given the coordinates of the vertices of a rectangle. We have to calculate the area of the figure formed by these points. To do that, we first draw a rough figure of the configuration. Then we find the length of the two unequal sides of the rectangle using the distance formula. And then finally, after calculating the lengths, we just multiply them to find the area of the given rectangular figure.

Formula Used:
1) We are going to use the distance formula:
Let there be two points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\]. Then 2) The distance between the points is:
\[{d_i} = \sqrt {{{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{x_2} - {x_1}} \right)}^2}} \]

Complete step by step solution:
Consider the following triangle with the given vertices and coordinates.

Now, let us calculate the lengths of any two adjacent (hence, unequal) sides of the rectangle \[ABCD\]. Let us calculate the lengths of sides \[AB\] and \[BC\].
First, let us find the length of side \[AB\]:
Coordinates between the two points are \[\left( {0, - 4} \right)\] and \[\left( { - 3,5} \right)\].
Hence, \[AB = \sqrt {{{\left( { - 3 - 0} \right)}^2} + {{\left( {5 - \left( { - 4} \right)} \right)}^2}} = \sqrt {9 + 81} = \sqrt {90} = 3\sqrt {10} \] units
Similarly, for the side \[BC\], coordinates are \[\left( { - 3,5} \right)\] and \[\left( {3,7} \right)\].
Hence, \[BC = \sqrt {{{\left( { - 3 - 3} \right)}^2} + {{\left( {5 - 7} \right)}^2}} = \sqrt {36 + 4} = \sqrt {40} = 2\sqrt {10} \] units
Thus, the area of rectangle \[ABCD\] is,

\[area = 3\sqrt {10} \times 2\sqrt {10} = 6 \times 10 = 60\] square units

Note:
For solving questions of such type, we first write what has been given to us. Then we write down what we have to find. Then we write the formula which connects the two things. It is advised that we always draw the figure of the configuration first, so that we know what thing we are dealing with. Then, we calculate the distance between the two points. This distance between any two given points is the length of the side joining two points.