Question

In figure, $M,N$ and $P$ are the midpoints of $AB$,$AC$ and $BC$ respectively. If $MN = 3cm$ ,$NP = 3.5cm$ and $MP = 2.5cm$, calculate the length of $BC,AB$ and $AC$.

Answer 1

Hint: In order to solve this question we have to apply the concept of mid-point theorem.
According to the mid-point theorem, the straight line joining the mid-points of two sides of a triangle is parallel to the third side and it is also half of the length of that side.
On further simplification we get the required answer.

Complete step-by-step solution:
It is given in the question that $M,N$ and $P$ are the mid-points of $AB$,$AC$ and $BC$ respectively.
We have to find the length of $BC,AB$ and $AC$.
Therefore by applying the concept of mid-point theorem we can write-
$MN\parallel BC$ and $MN = \dfrac{1}{2}BC$
It is given that the data $MN = 3cm$,$NP = 3.5cm$ and $MP = 2.5cm$,
So, we can write that-
$BC = 2MN$
$BC = 2MN = 2 \times 3 = 6cm$
Since the length of $MN = 3cm$
$\therefore $The length of $BC = 6cm$
Again from the diagram, since $M$ and $P$ are the mid-points of $AB$ and $BC$,
So we can write that-
$MP\parallel AC$ and $MP = \dfrac{1}{2}AC$
Hence we can write that-
$AC = 2MP$$ = 2 \times 2.5 = 5cm$
$\therefore $ The length of $AC = 5cm$
Also from the diagram, $N$ and $P$ are the mid-points of $AC$ and $BC$ respectively.
So we can write $NP\parallel AB$ and $NP = \dfrac{1}{2}AB$
Hence we can write that-
$2NP = AB$
So, $AB = 2NP = 2 \times 3.5 = 7cm$ since the length of $NP = 3.5cm$
$\therefore $ The length of $AB = 7cm$
Thus the length of $BC = 6cm$
Length of $AB = 7cm$
And length of $AC = 5cm$

Thus, the correct option is $A$.

Note: Mid-point means the middle point on any straight line which is equidistant from both the sides of the straight line.
According to the converse of the mid-point theorem, when a line is drawn through the midpoint of one side of a triangle then it is parallel to the third side and it bisects the third side.
Many of us get confused while applying the concept of mid-point theorem with the converse mid-point theorem. So keep in mind the statement of both the theorems to have a clear idea on each theorem and you can apply it whenever required.