Question

In the figure $BM = BN$ and $M$ is the midpoint of $AB$ and $N$ is the midpoint of$BC$. Then $AB = BC$.

A. True
B. False

Answer 1

Hint: Here we can see that $BM = BN$ and $BM + AM = AB{\text{ and }}BN + NC = BC$ in which we are also given that $M$ is the midpoint of $AB$ and $N$ is the midpoint of$BC$. Hence we can say that $AB = BC$

Complete Step by Step Solution:
Here we can see that $BM = BN$ as it is given in the problem.

Now we are also knowing that it is $BM + AM = AB{\text{ and }}BN + NC = BC$ as per the figure. Let
$
  BM + AM = AB - - - - \left( 1 \right) \\
  BN + NC = BC - - - - - \left( 2 \right) \\
 $
Now we are also given that $M$ is the midpoint of $AB$ and $N$ is the midpoint of$BC$.
As they are the mid-points on their respective line segments so we can equate the two halves that they are dividing. So we can say that:
$
  AM = BM - - - - (3) \\
  BN = NC - - - - \left( 4 \right) \\
 $
Now form the equation (1) we have got that $BM + AM = AB$
Let us substitute the value of $AM = BM$ we will get:
$
  BM + BM = AB \\
  2BM = AB - - - - (5) \\
 $
Now we can write the equation (4) as:
$BM = BN = NC$ as $BM = BN$ which is given in the problem.
Now we can write this value in equation (2) and we will get:
$
  BN + NC = BC \\
  BM + NC = BC - - - - \left( 6 \right) \\
 $
This is because $BM = BN$
As we have got that $BM = BN = NC$ and also that $AM = BM$so we can say that
$BM = BN = NC = AM$ so from this we can put $BM = NC$ in the above equation (6) and get:
$2BM = BC - - - - \left( 7 \right)$
So from equation (5) and (7) we get that:
$AB = BC$

Hence option A) is the correct option.

Note:
Here in these types of problems we can also be given the term centroid so we must know that centroid divides the line segment through which it is drawn in the ratio $2:1$.