Question

The vertices of the triangle are (2,-2), (4,2) and (-1,3). Find the equation of the median through (-1,3).

Answer 1

Hint: In this question first of all we will make proper diagram according to condition given in the question after it we will calculate coordinates of median. One coordinate is given and the other endpoint will be the midpoint of (2,-2) and (4,2), Then we will find the equation of line passing through the midpoint of (2,-2) and (4,2) and the point (-1,3).

Complete step by step answer:
To find the coordinates of D,

The two given points are (2,-2) and (4,2) and equate the points with \[({x_1},{y_1})\] and \[({x_2},{y_2})\] then
Coordinates of midpoint of two given points \[ = \dfrac{{{x_1} + {x_2}}}{2}\,\,,\,\dfrac{{{y_1} + {y_2}}}{2}\]
\[
   \Rightarrow \left( {\dfrac{{2 + 4}}{2},\;\dfrac{{ - 2 + 2}}{2}} \right) \\
   \Rightarrow (3,\,\,0) \\
\]
Hence the coordinates of point D is (3,0).

Now we are going to obtain the equation of straight line CD:
Here we have coordinates of both the points C(-1 , 3) and D(3 , 0).
Equation of the line passing through two given points \[({x_1},{y_1})\] and \[({x_2},{y_2})\]
\[ \Rightarrow (y - {y_1}) = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}(x - {x_1})\]
\[
\Rightarrow y - 3 = \dfrac{{0 - 3}}{{3 - ( - 1)}}(x - ( - 1)) \\
\Rightarrow y - 3 = \dfrac{{ - 3}}{4}(x + 1) \\
\Rightarrow 4y - 12 = - 3x - 3 \\
\Rightarrow 3x + 4y = 9 \\
\]
Hence, the equation of line is: \[3x + 4y = 9\].

Note: In this question we need to use the correct formula of finding the midpoint. And we need to use appropriate formulas to find the equation of line when two points are given.
Equation of the line passing through two given points \[({x_1},{y_1})\] and \[({x_2},{y_2})\]
\[ \Rightarrow(y - {y_1}) = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}(x - {x_1})\]
above formula can also be used as \[(y - {y_2}) = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}(x - {x_2})\].