Question

In a \[\Delta ABC,{\text{ }}AB = 4{\text{cm}}\] and \[AC{\text{ = 8cm}}\]. If $M$ is the mid-point of \[BC\] and \[AM = 3{\text{cm}}\], then the length of BC in cm is:
A) \[2\sqrt {26} \]
B) \[2\sqrt {31} \]
C) \[\sqrt {31} \]
D) \[\sqrt {26} \]

Answer 1

Hint: In the problem, we can use the Apollonius theorem to calculate the required value. Apollonius theorem states that the sum of squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side. Use it to find the length of $BM$which helps to approach the required result.

Complete step-by-step answer:
We have given a triangle ABC in which \[AB{\text{ = 4cm }}\], \[AC{\text{ = 8cm}}\] and ‘M’ is the midpoint of BC and \[AM{\text{ = 3cm}}\] and we have to find the length of BC in cm.
First of all we will draw a triangle that expresses all the given data.

We will use Apollonius theorem which gives that:
\[{\left| {AB} \right|^2} + {\left| {AC} \right|^2} = 2\left( {{{\left| {AM} \right|}^2} + {{\left| {BM} \right|}^2}} \right)\]
Let us assume that \[BM = x{\text{ }}cm\] and put all the values in the Apollonius theorem equation, we will get:
\[{4^2} + {8^2} = 2\left( {{3^2} + {x^2}} \right)\]
Simplify the above equation for the value of$x$.
$
  16 + 64 = 18 + 2{x^2} \\
  80 - 18 = 2{x^2} \\
  62 = 2{x^2} \\
  \dfrac{{62}}{2} = {x^2} \\
  31 = {x^2} \\
  \sqrt {31} = x \\
$
The obtained value of $x$ is $\sqrt {31} $, as assumes the length of the side $BM$ is $\sqrt {31} $cm.
It is given that M is the midpoint of the side $BC$, it means that the point M divides the line BC in two equal parts. That is,
$BM = CM$
It is also clear that BC can be express as:
$BC = BM + CM$
As BM is equal to CM, then we obtained:
$BC = 2BM$
Now, substitute the obtained value of BM in the equation of BC to find BC.
\[BC = 2\sqrt {31} cm\]
SO, the obtained length of the side of the triangle $BC$ is $2\sqrt {31} $cm.

Hence, the correct option is (B).

Note: It can be seen in the problem that we have to find the relation between the median and the side lengths of the triangle and the best fitting theorem is Apollonius’s theorem, which gives you relation in the median of the triangle and the side lengths of the triangle.