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:

Answer: Option D

According to question,

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


$$\begin{array}{rl}& \therefore \frac{BC}{AB}=\frac{AB}{BD}\\ & \therefore \frac{BC}{\sqrt{52}}=\frac{\sqrt{52}}{4}\\ & BC=\frac{52}{4}\\ & BC=13cm\end{array}$$

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