In a triangle ABC, ∠BAC = 90° and AD is perpendicular to BC. If AD = 6 cm and BD = 4 cm then the length of BC is:

Solution(By Examveda Team)

According to question,

Given: BAC is a right angle triangle
AD ⊥ BC
AD = 6 cm
BD = 4 cm
BC = ?
In ΔBAD
$$\eqalign{ & AB = \sqrt {B{D^2} + A{D^2}} \cr & AB = \sqrt {{4^2} + {6^2}} \cr & AB = \sqrt {52} \,cm \cr} $$
ΔBAC ∼ ΔBDA


$$\eqalign{ & \therefore \frac{{BC}}{{AB}} = \frac{{AB}}{{BD}} \cr & \therefore \frac{{BC}}{{\sqrt {52} }} = \frac{{\sqrt {52} }}{4} \cr & BC = \frac{{52}}{4} \cr & BC = 13\,cm \cr} $$

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Alternate :
$$\eqalign{ & A{B^2} = BD.BC \cr & {\left( {\sqrt {B{D^2} + A{D^2}} } \right)^2} = BD.BC \cr & {\left( {\sqrt {{4^2} + {6^2}} } \right)^2} = 4.BC \cr & \frac{{52}}{4} = BC, \cr & \therefore BC = 13\,cm \cr} $$