In the given figure, ∠A = 90° and AD ⊥ BC. If BD = 2 cm and CD = 8 cm, find AD.

Given ∠A = 90°,

or, ∠BAD + ∠DAC = 90° …..(i)

Now, consider △ADC

∠ADC = 90°

or, ∠DCA + ∠DAC = 90° …..(ii)

From equation (i) and equation (ii)

We have,

∠BAD + ∠DAC = ∠DCA + ∠DAC

∠BAD = ∠DCA …..(iii)

So, from △BDA and △ADC

∠BDA = ∠ADC = 90°
∠BAD = ∠DCA [From equation (iii)]

So, by AA rule of similarity △BDA ~ △ADC.

Since, corresponding sides of similar triangles are proportional,

$\therefore \dfrac{BD}{AD} = \dfrac{AD}{DC} = \dfrac{AB}{AC} \\[1em] \text{Considering, } \dfrac{BD}{AD} = \dfrac{AD}{DC} \\[1em] AD^2 = BD \times CD \\[1em] AD^2 = 2 \times 8 \\[1em] AD^2 = 16 \\[1em] AD^2 - 16 = 0 \\[1em] AD^2 - 4^2 = 0 \\[1em] (AD - 4)(AD + 4) = 0 \\[1em] AD - 4 = 0 \text{ and } AD + 4 = 0 \\[1em] AD = 4 \text{ and } AD = -4.$

Since, length cannot be negative hence, AD ≠ -4.

Hence, the length of AD = 4 cm.