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

• A. $45°$
• B. $60°$
• C. $90°$
• D. $30°$

Answer 1

The correct option is C

$90°$

Explanation for the correct option:

Steps 1: Given data

$ABCD$ is a quadrilateral

Equation of $AB$ is $x+2y=3.....\left(1\right)$

Equation of $BC$ is $x=1.....\left(2\right)$

Equation of $CD$ is $x-3y=4.....\left(3\right)$

Equation of $DA$ is $5x+y+12=0.....\left(4\right)$

Step 2: Find coordinates of the vertices of the quadrilateral

From $\left(1\right)$ and $\left(2\right)$

$B=(1,1)$

From $\left(2\right)$ and $\left(3\right)$

$C=(1,-1)$

From $\left(3\right)$ and $\left(4\right)$

$D=(-2,-2)$

From $\left(4\right)$ and $\left(1\right)$

$A=(-3,3)$

Step 3: Find the slope of the diagonals

Slope of $AC$,

$m_1=\frac{(3+1)}{(-3-1)}=-1$

Slope of $BD$,

$m_2=\frac{\left(1+2\right)}{(1+2)}=1$

Multiplying the slopes,

$m_1\times m_2=-1$

$\therefore$$AC$ and $BD$ are perpendicular.

So, the angle between the diagonals is $90°$.

Hence, option (C) is correct.