In the given figure, ∠B = 90°, XY // BC, AB = 12 cm, AY = 8 cm and AX : XB = 1 : 2 = AY : YC. Find the lengths of AC and BC.

Given,

AX : XB = 1 : 2

Let AX = x and XB = 2x.

From figure,

⇒ AX + XB = AB

⇒ x + 2x = 12

⇒ 3x = 12

⇒ x = $\dfrac{12}{3}$ = 4 cm.

⇒ AX = x = 4 cm and XB = 2x = 2(4) = 8 cm.

Given,

AY : YC = 1 : 2

$\Rightarrow \dfrac{AY}{YC} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{8}{YC} = \dfrac{1}{2} \\[1em] \Rightarrow YC = 8 \times 2 = 16.$

From figure,

⇒ AC = AY + YC = 8 + 16 = 24 cm.

In right-angled triangle ABC,

By pythagoras theorem,

⇒ (Hypotenuse)² = (Perpendicular)² + Base²

⇒ AC² = AB² + BC²

⇒ 24² = 12² + BC²

⇒ 576 = 144 + BC²

⇒ BC² = 576 - 144

⇒ BC² = 432

⇒ BC = $\sqrt{432} = 12\sqrt{3}$ = 20.78 cm.

Hence, AC = 24 cm and BC = 20.78 cm.