In a $$\triangle$$ABC, $$\angle$$BAC = $$90^\circ$$ and AD is perpendicular to BC where D is a point on BC. If BD = 4 cm and CD = 5 cm,then the length of AD is equal to:

In $$\triangle$$ABC,

$$\angle$$A = 90$$^\circ$$

$$\Rightarrow$$  $$\angle$$B = 90$$^\circ$$ - $$\angle$$C

AD is perpendicular to BC

$$\Rightarrow$$  $$\angle$$ADC = 90$$^\circ$$

In $$\triangle$$ADC,

$$\angle$$ADC = 90$$^\circ$$

$$\Rightarrow$$  $$\angle$$DAC = 90$$^\circ$$ - $$\angle$$C

In $$\triangle$$ADB and $$\triangle$$CDA,

$$\angle$$B = $$\angle$$DAC = 90$$^\circ$$ - $$\angle$$C

$$\angle$$BDA = $$\angle$$ADC = 90$$^\circ$$

Two angles are equal for both the triangles, $$\triangle$$ADB is similar to $$\triangle$$CDA

Ratio of respective sides are equal in both the triangles

$$\Rightarrow$$  $$\frac{BD}{AD}=\frac{AD}{CD}$$

$$\Rightarrow$$  AD$$^2$$ = BD.CD

$$\Rightarrow$$  AD$$^2$$ = 4 x 5

$$\Rightarrow$$  AD$$^2$$ =  20

$$\Rightarrow$$  AD = $$2 \sqrt 5$$ cm

Hence, the correct answer is Option B