Question

The vertices of a triangle are A (-1,3) ,B (1,-1) and C (5,1). Find the length of the median through the vertex C.

Answer 1

Hint: Find the midpoint of straight-line BC, then use distance formula to find the length of median passing through A.

Complete step-by-step answer:
Let q be the distance between two points (a, b) and (c, d) is given by:
$q=\sqrt{{{\left( a-c \right)}^{2}}+{{\left( b-d \right)}^{2}}}$
First, we need to find the mid-point of the straight-line AB. For co-ordinates of midpoint, we need the average of the values of $x$ - coordinates and $y$ - coordinates of point B and point A. Let the mid-point be named as D.
Co-ordinates of D = Average of coordinates of B, A
Let $x$ - coordinate of D = p,
       $y$ - coordinate of D =q
From the given points we can say,
$x$ - coordinate of B = 1, $y$ - coordinate of B = - 1
$x$ - coordinate of A = -1, $y$ - coordinate of A = 3
p = Average of $x$ - coordinates of B,A
\[p=\left( \dfrac{x-\text{coordinate of B + }x\text{ coordinate of A }}{2} \right)\]
By substituting the values, we get:
\[p=\dfrac{1-1}{2}\]
p = 0
Similarly
q = Average of $y$ - coordinates of point B and point C
$q=\dfrac{3+\left( -1 \right)}{2}=\dfrac{3-1}{2}=\dfrac{2}{2}$
q = 1
So, mid-point of line AB = (0, 1)
$\Rightarrow D=\left( 0,1 \right)$

In the above figure we see the median CD.
Now, we need length of median
A median is a line which passes through a vertex to the mid-point of the side opposite to the particular vertex.
We need the length of the median through the vertex C.
By applying distance formula from the vertex C to the mid-point of side AB, we get the length of median through C.
We know, distance formula:
If distance between two points (a, b), (c, d) is q then:
$q=\sqrt{{{\left( c-a \right)}^{2}}+{{\left( d-b \right)}^{2}}}$
Here, we need distance between point C (5, 1) and the mid-point of side AB: D(0, 1)
From above formula let:
AD = q, a = 5, b = 1, c = 0, d = 1
By substituting above values in the equation we get
$\begin{align}
  & q=\sqrt{{{\left( 5-0 \right)}^{2}}+{{\left( 1-1 \right)}^{2}}}=\sqrt{25} \\
 & q=5 \\
 & \Rightarrow CD=5 \\
\end{align}$
CD is the median through C.
Therefore, the length of median through C is 5 units.

Note: While calculating the distance formula, take the sign of co-ordinates into consideration. If not, you may lead to the wrong answer.