In triangle ABC, ∠B = 90° and tan A = 0.75. If AC = 30 cm, find the lengths of AB and BC.

Given:

tan A = 0.75 = $\dfrac{75}{100}$

tan A = $\dfrac{3}{4}$

$\text{tan A} = \dfrac{Perpendicular}{Base} = \dfrac{3}{4}$

∴ If length of AB = 4x unit, length of BC = 3x unit.

In Δ ABC,

⇒ AC² = AB² + BC² (∵ AC is hypotenuse)

⇒ AC² = (4x)² + (3x)²

⇒ AC² = 16x² + 9x²

⇒ AC² = 25x²

⇒ AC = $\sqrt{25\text{x}^2}$

⇒ AC = 5x

It is given that AC = 30 cm.

So, 5x = 30

x = $\dfrac{30}{5} = 6$

Therefore, AB = 4x = 4 x 6 cm = 24 cm and BC = 3x = 3 x 6 cm = 18 cm.

Hence, AB = 24 cm and BC = 18 cm.