Question

M, N and P are the mid-points of $ AB,AC $ and $ BC $ respectively. If $ MN=3cm,NP=3.5cm $ and $ MP=2.5cm $ , $ BC,AB $ and $ AC $ ?

Answer 1

Hint: We will use the mid-point theorem to find out the dimensions of the side of the triangle asked. Which says that triangles formed by joining the mid-points of a triangle have length half of the actual triangle.

Complete step by step answer:
Moving ahead with the question in step-wise manner;
M, N and P are the mid-points of $ \Delta ABC $ . So according to the mid-point theorem of triangle states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.” As taking example from figure 1, the line joined by the midpoints of $ AB,AC $ which is line $ MN $ will be parallel and half of the third side ( $ BC $ ).
As all three points M, N and P are the mid-points of $ \Delta ABC $ .By using the mid-point theorem we can say that the line joining these mid-points will be parallel and half to their third side. So by comparing we can say that the line $ MN,NP $ and $ MP $ are parallel and half to the line $ BC,AB,AC $ respectively.
So we can write
 $ \begin{align}
  & MN=\dfrac{BC}{2}, \\
 & NP=\dfrac{AB}{2} \\
 & and \\
 & MP=\dfrac{AC}{2} \\
\end{align} $
As we know that $ MN=3cm,NP=3.5cm $ and $ MP=2.5cm $ ,put these values in above relation to find the dimensions of sides of triangle;
So for BC we know that;
 $ MN=\dfrac{BC}{2} $
Put $ MN=3cm $ so we will get;
 $ \begin{align}
  & 3=\dfrac{BC}{2} \\
 & BC=6cm \\
\end{align} $
Similarly for AB we know that;
 $ NP=\dfrac{AB}{2} $
Put $ NP=3.5cm $ so we will get;
 $ \begin{align}
  & 3.5=\dfrac{AB}{2} \\
 & AB=7cm \\
\end{align} $
Similarly for AC we know that;
 $ MP=\dfrac{AC}{2} $
Put $ MP=2.5cm $ so we will get;
 $ \begin{align}
  & 2.5=\dfrac{AC}{2} \\
 & AC=5cm \\
\end{align} $
Hence $ BC=6cm,AB=7cm $ and $ AC=5cm $

Note: According to the mid-point theorem not only we have the relation between the line joining the midpoints and sides of the triangle. Moreover the triangles formed by joining the midpoints of triangle are congruent to the main triangle.