Question

Plot the points \[A(1,2),B( - 4,2),C( - 4, - 1)\] and \[D(1, - 1)\]. What kind of quadrilateral is ABCD? Also find the area of the quadrilateral ABCD.

Answer 1

Hint: According to given in the question we have to determine the area of the quadrilateral ABCD and we have to plot the points of the quadrilateral ABCD. So, first of all we have to draw the plot for the given quadrilateral ABCD having points \[A(1,2),B( - 4,2),C( - 4, - 1)\] and \[D(1, - 1)\] which can be obtained with the help of the quadrant graph in which we have to draw both x-axis and y-axis and then we have to check for the points and divide the axis.
Now, we have to check that the obtained quadrilateral is a square, rectangle or a parallelogram.
Now, we have to find the distance between the points AB and AD which can be determined with the help of the formula to find the distance between the two points which is as below:

Formula used:
 $ \Rightarrow \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} ...................(A)$
Hence, with the help of the formula (A) above, we can determine the lengths of the given quadrilateral AB and AD.
Now, we have to determine the area of the given quadrilateral ABCD which can be determine with the help of the formula to determine the area as mentioned below:
$ \Rightarrow l \times b.............(B)$
So, with the help of the formula (B) as mentioned above where, l is the length and b is the breadth of the quadrilateral ABCD hence, on substituting all the values in the formula above, we can obtain the required area.

Complete step-by-step answer:
Step 1: First of all we have to plot all the points of the given quadrilateral ABCD, having points \[A(1,2),B( - 4,2),C( - 4, - 1)\] and \[D(1, - 1)\] which can be obtained with the help of the quadrant graph in which we have to draw both x-axis and y-axis and then we have to check for the points and divide the axis as mentioned in the solution hint. Hence,

Step 2: Now, as from the solution step 1 we have determined the plot for the given points of the quadrilateral and we can say that the obtained quadrilateral is a rectangle because its opposite sides are parallel and equal to each other and the angle between all the sides is a right angle.
Step 3: Now, we have to determine the distance between the points or we can say the length of the lines of the triangle ABCD which are AB and AD with the help of the formula (A) as mentioned in the solution hint. Hence,
$
   \Rightarrow AB = \sqrt {{{( - 4 - 1)}^2} + {{(2 - 2)}^2}} \\
   \Rightarrow AB = \sqrt {{{( - 5)}^2} + {0^2}} \\
   \Rightarrow AB = \sqrt {25} \\
   \Rightarrow AB = 5 \\
 $
Step 4: Now, same as the solution step 3 we have to determine the distance between the points A and D or we have to find the length of the line AD with the help of the formula (A) as mentioned in the solution hint. Hence,
$
   \Rightarrow AD = \sqrt {{{(1 - 1)}^2} + {{( - 1 - 2)}^2}} \\
   \Rightarrow AD = \sqrt {{{(0)}^2} + {{( - 3)}^2}} \\
   \Rightarrow AD = \sqrt 9 \\
   \Rightarrow AD = 3 \\
 $
Step 5: Now, we have to find the area of the given triangle ABCD with the help of the formula (B) which is as mentioned in the solution hint. Hence, on substituting all the values in the formula (B),
$ \Rightarrow $Area of ABCD
$
   = 5 \times 3 \\
   = 15uni{t^2} \\
 $

Hence, with the help of the formula (A) and (B) we have determined the area of the given quadrilateral ABCD which is $15uni{t^2}$.

Note:
To obtain the area of the given quadrilateral it is necessary that we have to determine the given quadrilateral is a square, rectangle, rhombus or a parallelogram which can be determine by plotting all the given points for the quadrilateral.To determine the area it is necessary that we have to determine the distance between the points A and B and same as A and D which can be determine with the help of the formula to find the distance between the two points.