Question

If the points $A(1,2),B(2,3)C(a,2)$ and $D(4,3)$ form a parallelogram, find the value of $a$ and height of the parallelogram taking $AB$ as base.

• A. $a=1,h=2$
• B. $a=1,h=1$
• C. $a=3,h=1$
• D. $a=2,h=3$

Answer 1

Answer: (C)

$a=3,h=1$

Given that $ABCD$ is a parallelogram with $A(x_1,y_1)=(1,2),B(x_2,y_2)=(2,3)$

$C(x_3,y_3)=(a,2)$ and $D(x_4,y_4)=(4,3)$.

The ordinates of $A(x_1,y_1)=(1,2)$ & $B(x_2,y_2)=(2,3)$ are same which is $3$.

$\therefore AB\parallel x$-axis............(i) and

$AB=CD=$ difference between the abscissa

$=(x_2-x_1)=(2-1)$ units $=1$ unit ...........(ii)

Also $AB=x_3-x_1=a-2$ .......(iii)

$\therefore$ from (i) & (ii) $a-2=1$

$\Rightarrow a=3$

Again $AB\parallel CD$ ....(opposite sides of a parallelogram.)

So, $AB\parallel CD\parallel x$-axis.

$\therefore$ the height $h=mn$ between $AB$ and $CD$ will be the difference between the respective ordinates i.e. $d=y_4-y_{1}ord=y_3-y_2$

So, $h=mn=(3-2)$ unit $=1$ unit or $|2-3|$ unit $=1$ unit