Question

Equation of median through vertex B of $\mathrm{\Delta }ABC$ where A (0, 0), B (0,1) and C (1,0) is
(a)y + 2x = 1
(b)2y + 2x = 1
(c)x + y = 1
(d)3x + 2y = 2

Answer 1

Hint: First of all find the centroid of the triangle because the median of the triangle from any point passes through the centroid. As we have to find an equation of a median passing through point B so we have a point B and the centroid. So, we can write the equation of a line if two points are given.

Complete step-by-step answer:
The triangle ABC and the median passing through B are shown below:

Firstly, we are going to find the centroid of the$\mathrm{\Delta }ABC$. So, the formula of the centroid of$\mathrm{\Delta }ABC$is shown below:
$D(x,y)=({\displaystyle \frac{x_1+x_2+x_3}{3}},{\displaystyle \frac{y_1+y_2+y_3}{3}})$
Here x1, x2, x3, y1, y2, y3 are the x and y coordinates of A, B and C respectively. A(x1, y1), B(x2, y2), C(x3, y3).
Now, putting the values of A, B and C in the formula of centroid will give:
$\begin{array}{rl}& D(x,y)=({\displaystyle \frac{0+0+1}{3}},{\displaystyle \frac{0+1+0}{3}})\\ & \Rightarrow D(x,y)=({\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{3}})\end{array}$
Now, we know the centroid of$\mathrm{\Delta }ABC$ and the point B is already given in the question so we can write the equation of a line passing through the median.
We know point B (0, 1) and centroid $D({\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{3}})$. So, for finding the equation of a line passing through these points we will first find the slope of the line.
The formula for slope of the line passing through two points P(x1, y1) and Q(x2, y2) is as follows:
$m={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$
The point to be noted here is that these x1, x2, y1, y2 are not the ones that we have described above for the centroid.
Now, we are going to find the slope of the line passing through median:
$\begin{array}{rl}& m={\displaystyle \frac{{\displaystyle \frac{1}{3}}-1}{{\displaystyle \frac{1}{3}}-0}}\\ & \Rightarrow m=-2\end{array}$
Now, we know the slope and take any one point from B and D. We are taking point B and we are writing the equation of a line passing through B and D.
$y-y_1=m(x-x_1)$
In the above equation, the value of m =-2 and (x1, y1) is the point B (0, 1).
$\begin{array}{rl}& y-1=-2(x-0)\\ & \Rightarrow y-1=-2x\\ & \Rightarrow 2x+y=1\end{array}$
Hence, the equation of the median passing through B is 2x + y = 1.
Hence, the correct option is (a).

Note: We can eliminate the options by putting the value of x and y of the coordinates of point B and the centroid of the triangle. The equation which satisfies both the centroid and the point B is the correct equation.