Question

A triangle ABC in which $ AB = AC $ , M is a point on AB and N is a point on AC such that if $ BM = CN $ then $ AM = AN $
A. True
B. False

Answer 1

Hint: Here in the given question, we have to check whether the given condition of the triangle is true or false using the given data. First, we have to construct a triangle using the given measurements in the question and next by midpoints divide the sides of a triangle and by subtraction and on further simplification, we get the required solution.

Complete step by step solution:

In a given \[\Delta \,ABC\] side $ AB = AC $ so it’s a isosceles triangle,
Draw a line \[MN\] from a point \[M\]to \[N\], where \[M\] is a midpoint on \[AB\] and \[N\]is a midpoint on AC.
Now, we have to check whether if the length BM is equal to the length CN then the length AM is equal to the length AN.
i.e., if $ BM = CN $ then $ AM = AN $
Since M is a point on the side AB,
 $ \Rightarrow \,\,AB = AM + BM $ …. (i)
And since N is the point on the side AC,
 $ \Rightarrow \,\,AC = AN + CN $ …. (ii)
Now, subtract equation (ii) from equation (i), Then we get that $ \Rightarrow \,\,AB - AC = AM + BM - (AN + CN) $ … (iii)
It is given that $ AB = AC $ . Therefore, substitute $ AB = AC $ in equation (iii). Then we get
  $ \Rightarrow \,\,AC - AC = AM + BM - (AN + CN) $ .
 $ \Rightarrow AM + BM - (AN + CN) = 0 $ … (iv)
Now, substitute $ BM = CN $ in equation (iv). This gives us that
 $ \Rightarrow \,\,AM + CN - (AN + CN) = 0 $
 $ \Rightarrow AM + CN - AN - CN $
 $ \Rightarrow AM - AN = 0 $
 $ \Rightarrow AM = AN $ .
Therefore, in the given triangle if $ BM = CN $ then $ AM = AN $ .
Hence, the given statement is true.
With this, the correct option is A.
So, the correct answer is “Option A”.

Note: Remember, when the midpoint of line divides the line equally. And this type of problems also solved by using the properties of similarities of two triangles i.e., when two triangles are similar and congruent then the corresponding sides of the two triangles should be equal.