Question

The sides $AB,BC,CD$ and $DA$ of a quadrilateral are $x + 2y = 3,x = 1,x - 3y = 4,5x + y + 12 = 0$ respectively. The angle between diagonals $AC$ and $BD$ is
1. ${45^ \circ }$
2. ${60^ \circ }$
3. ${90^ \circ }$
4. ${30^ \circ }$

Answer 1

Hint: In this question, we are given the equations of each side of the quadrilateral, and we have to find the angle between both diagonals. The first step is to find the coordinates of the quadrilateral using substitution, elimination, or graphical method (using any one method). Then calculate the slopes using the slope formula and you’ll get the condition that the product of slopes is equal to $ - 1$. It is only possible when the lines are perpendicular to each other.

Formula Used:
Slope formula –
$m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ , where $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ are the coordinates of line
If two lines are perpendicular it means the product of their slopes should be equal to $ - 1$
i.e., ${m_1}{m_2} = - 1$ where ${m_1},{m_2}$ are the slopes of two lines.

Complete step by step Solution:
Given equations of the sides of a quadrilateral are
$x + 2y = 3 - - - - - \left( 1 \right)$
$x = 1 - - - - - \left( 2 \right)$
$x - 3y = 4 - - - - - \left( 3 \right)$
$5x + y + 12 = 0 - - - - - \left( 4 \right)$
Now, to find the coordinates of the quadrilaterals we are solving the equations by making pairs for each coordinate,
From equation (1) and (2), we get $x = 1,y = 1$
Therefore, the coordinates of $B = \left( {1,1} \right)$
From equation (2) and (3), we get $x = 1,y = - 1$
Therefore, the coordinates of $C = \left( {1, - 1} \right)$
From equation (3) and (4), we get $x = - 2,y = - 2$
Therefore, the coordinates of $D = \left( { - 2, - 2} \right)$
From equation (1) and (4), we get $x = - 3,y = 3$
Therefore, the coordinates of $A = \left( { - 3,3} \right)$
The figure is attached below,

Figure 1- Image: A quadrilateral ABCD, whose coordinates are A (-3,3), B(1,1), C(1,-1) and D(-2,-2)
Now, we’ll find the slope of diagonals using the slope formula
$m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ , where $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ are the coordinates of line
Slope of diagonal $AC = \dfrac{{ - 1 - 3}}{{1 - \left( { - 3} \right)}} = - 1$
Slope of diagonal $BD = \dfrac{{ - 2 - 1}}{{ - 2 - 1}} = 1$
Here, the product of the slope of both the diagonals is equal to $ - 1$
It implies that, Both the diagonals are perpendicular to each other
Therefore, the angle between diagonals $AC$ and $BD$ is ${90^ \circ }$.

Hence, the correct option is 3.

Note: The key concept involved in solving this problem is a good knowledge of methods to find unknown terms. Students must know that there are three methods to find the unknown terms: Elimination method, Substitution method, and graphical method. Here, we have applied the elimination method, which is basically the process of eliminating one of the variables in a system of linear equations by just using any of the operation addition or subtraction methods in conjunction with the variable coefficient multiplication or division. You can solve using any of the methods that will also be correct.