The measurement of h is :

1. $\sqrt{8 \times 6}$ cm

2. $\sqrt{\dfrac{4}{3}}$ cm

3. (8 - 6) cm

4. $2\sqrt{2}$ cm

We know that,

tan θ = $\dfrac{\text{Perpendicular}}{\text{Base}}$

From figure,

In △ ABC,

⇒ tan 50° = $\dfrac{AB}{BC}$

⇒ tan 50° = $\dfrac{h}{6}$

⇒ h = 6 tan 50°

⇒ h = 6 tan (90° - 40°)

⇒ h = 6 cot 40° ……..(1)

In △ ABD,

⇒ tan 40° = $\dfrac{AB}{BD}$

⇒ tan 40° = $\dfrac{h}{8}$

⇒ h = 8 tan 40° ……..(2)

Multiplying equation (1) and (2), we get :

⇒ h × h = 6 cot 40° × 8 tan 40°

⇒ h² = 8 × 6 × cot 40° $\times \dfrac{1}{\text{cot 40°}}$

⇒ h² = 8 × 6

⇒ h = $\sqrt{8 \times 6}$.

Hence, Option 1 is the correct option.