Question

In the figure, ABCD is a parallelogram.
a) Write the coordinates of D.
b) What is the height of this parallelogram?
c) Find the perimeter and area of it.

Answer 1

Hint: Here we will find the coordinates of D, height, perimeter and area of the parallelogram by considering the geometry concepts and properties of parallelogram.

Complete step-by-step answer:
a) Coordinates of D
Distance between A and B
$\Rightarrow \sqrt{{(4-0)}^2+{(2-2)}^2}=\sqrt{16}=4$
Since the opposite sides of a parallelogram are equal, the distance between C and D is also 4. Also, the y-coordinates of A and B are equal, the sides AB is parallel to the x-axis. And since ABCD is a parallelogram, CD is also parallel to the x-axis. So the y-coordinates of C and D are also equal.
Thus, the x-coordinate of $D=8-4=4$
Hence, the coordinates of ’D’ are (4, 5).

b) Height of the parallelogram
The x-coordinates of D and B are equal. So the line BD is parallel to the y-axis or BD is perpendicular to the x-axis.
So, height of the parallelogram $=DB=\left|y_1-y_2\right|=\left|5-2\right|=3$

c) Perimeter and area of the parallelogram
Find the distance between B and C
$$\begin{gathered} BC=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2} \\ BC=\sqrt{{(8-4)}^2+{(5-2)}^2} \\ BC=\sqrt{{(4)}^2+{(3)}^2} \\ BC=\sqrt{16+9} \\ BC=\sqrt{25}=5 \end{gathered}$$
$AD=BC=5\text{ units}$


Perimeter=$2(4+5)=2\times 9=18$units.
Area=Base x Height
Area$=4\times 3=12$ sq .units

Note: In this type of question we have to apply the relationship between the parallel line and perpendicular line and use the various formulas like distance formula to arrive at the solution.