Question

Hey,

In a ∆ABC, M and N are points on the base BC such that angle MAB =angle BCA and CAN = angle ABC. If AM 2 cm, BM = 3 cm and AN =6 cm then NC =​​

Answer 1

Step-by-step explanation:

In ΔABC, M and N are points on the sides AB and AC respectively.

If ∠B=∠C,we know that, sides opposite to equal angles are equal AB=AC

BM=CN ___ (1)

AB−BM=AC−CN

⇒AM=AN ___ (2)

From ΔABC 

Dividing (2) by (1), we get

MBAM=NCAN 

By converse of basic proportionality theorem, 

  MN∣∣BN

Answer 2

Given: In a ∆ABC, M and N are points on the base BC such that angle MAB =angle BCA and CAN = angle ABC. If AM 2 cm, BM = 3 cm and AN =6 cm.

To find: NC

Solution:

The triangles ABM and ACN are similar as two of their angles are equal and the opposite side of both the triangles are in proportion. So, they are similar by the postulate ASA (Angle-Side-Angle). Thus, the sides that are in proportion can be written as follows.

$\frac{AB}{AC} = \frac{BM}{AN} = \frac{AM}{NC}$

$\frac{AB}{AC} = \frac{3}{6} = \frac{2}{NC}$

$NC = 4 cm$

The measure of NC is calculated to be 4 cm in length.

Therefore, the length of NC is 4 cm.