Question

What is the y intercept line that is parallel to $y=3x,$ and which bisects the area of rectangle with corners at $\left( 0,0 \right),\left( 4,0 \right),\left( 4,2 \right)\,\,and\,\left( 0,2 \right)$ ?
$\begin{align}
  & A.-7 \\
 & B.-6 \\
 & C.-5 \\
 & D-4 \\
\end{align}$

Answer 1

Hint: In this question, first find the midpoint of the rectangle by using distance formula.
Then use the line equation with the value of slope and midpoint and at last compare the obtained equation with the general equation of line.

Complete step-by-step answer:
To solve this question, first we have to find the mid-point of the rectangle by using distance formula and let us draw one rectangle with the help of given vertices

Midpoint of rectangle =$(\dfrac{0+4}{2},\dfrac{0+2}{2})=(2,1)$
By using the given coordinates of the rectangle in distance formula.
Now we have a line equation $y=3x,$
Then, slope of the above line will be ${{m}_{1}}=3$
Therefore slope of the line parallel to the given line will be${{m}_{2}}=3$
Now equation of the line with slope $3$ and passing through the point $(2,1)$ is
$\Rightarrow$ $y-{{y}_{1}}=m(x-{{x}_{1}})$
On putting all values in the above equation, we get
$\Rightarrow$ $y-1=3(x-2)$
On transferring number one side and rearrange the obtained equation in the lin equation, we get
$\Rightarrow$ $y=3x-5$
On comparing the obtained equation with general equation, we get
$\Rightarrow$ $y=mx+c$
Intercept c =$-5$
Hence option C is the correct .

Additional information:
Distance formula for midpoint-The point that is at the same distance from two points on a line is called midpoint.
We can calculate midpoint by using this formula
$\left( \dfrac{{{x}_{1}}+{{x}_{2}}}{2} \right),\left( \dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)$

Note: In this problem we were using two different concepts , first distance formula and second line equation, so that students should be aware that according to the question requirement we have to apply different formulas or concepts.