Question

The length of the median from the vertex A of a triangle whose vertices are A\[( - 1,3)\], B\[(1, - 1)\] and C \[(5,1)\] is
A.\[5\]
B.\[4\]
C.\[1\]
D.\[3\]

Answer 1

Hint: We will use the mid-point formula to calculate the other vertex of the median and then use the distance formula to calculate the length of the median. A median is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side.

Complete step-by-step answer:
From the given question, we know
The coordinates of A\[ = ( - 1,3)\]
The coordinates of B\[ = (1, - 1)\]
The coordinates of C\[ = (5,1)\]
The median drawn through vertex A will meet BC at the midpoint of BC.
We know that the midpoint formula is \[\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)\].
Using the above formula, we get the coordinates of midpoint of BC
\[
   = \left( {\dfrac{{1 + 5}}{2},\dfrac{{ - 1 + 1}}{2}} \right) \\
   = (3,0) \\
\]
Now, the length of the median through vertex A = The distance between \[( - 1,3)\]and \[(3,0)\].
We know that the distance formula is \[\sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} \]to find the distance between any two points.
So, now using the above formula, we get
the length of the median through vertex A
\[
  \sqrt {{{(3 - ( - 1))}^2} + {{(0 - 3)}^2}} \\
   = \sqrt {{{(3 + 1)}^2} + {{( - 3)}^2}} \\
   = \sqrt {{4^2} + {{( - 3)}^2}} \\
   = \sqrt {16 + 9} \\
   = \sqrt {25} = \pm 5 \\
\]
Either the length is \[5\] or \[ - 5\].
The length cannot be negative, therefore the length of the median is \[5\].
Thus, the answer is option A.
Note: The median of a scalene triangle only bisects the opposite side of the vertex and not the whole triangle. The types of triangles that can be bisected by a median are an isosceles triangle and an equilateral triangle.