In △ABC, m∠ABC=40°, BL (L∈ AC ) is the angle bisector of ∠B.
Point M∈ AB so that LM ⊥ AB and N∈ BC so that LN ⊥ BC .
m∠CLN=3m∠ALM
Let m∠ALM = x so m∠CLN = 3x
According to the diagram,
m∠A = 90 - x and m∠c= 90 - 3x
As we know that sum of all the angles of a triangle is 180,
m∠A + m∠B + m∠C = 180
(90 - x) + 40 + (90 - 3x) = 180
220 - 4x = 180
-4x = -40
x = 10
So m∠A = 90 - x = 90 - 10 =80
m∠C = 90 - 3x = 90 - 30 = 60
In ΔCLN, ∠N= 90°, ∠C = 60° and ∠CLN = 30°
So by 30 -60-90 theorem
Side opposite to 90° = 1/2( side opposite to 30°)
CN = 1/2 (CL)
