In $\triangle$ ABC, $\angle C = 90^\circ$. M and N are the mid-points of sides AB and AC,respectively. CM and BN intersect each other at D and $\angle BDC = 90^\circ$. If BC = 8 cm, then the length of BN is:
Answer & explanation
Correct answer: option 3
As BN and CM are medians of triangle ABC, D is centroid of the triangle.
So, D will divide BN in the ratio 2 : 1
= BD : DN = 2 : 1
Let BD = 2x and DN = x
= BN = 3x
Also in right angled triangle CNB, CD perpendicular to BN
= \( {BC }^{ 2} \) = BD x BN
= \( {8}^{ 2} \) = 2x x 3x
= 64 = \( {6x }^{ 2} \)
= x = \(\frac{4√6}{3}\)
= BN = 3x = 3 x (\(\frac{4√6}{3}\)) = 4√6
Therefore, BD is 4√6 cm.