Question

In the given figure, ∠B = 90^°, XY || BC, AB = 12 cm, AY = 8cm and AX : XB = 1 : 2 = AY : YC.

Find the lengths of AC and BC.

Answer 1

Given that AX : XB = 1 : 2 = AY : YC.

Let x be the common multiple for which this proportion gets satisfied.

So, AX = 1x and XB = 2x

AX + XB = 1x + 2x = 3x

⇒ AB = 3x    .….(A - X - B)

⇒ 12 = 3x

⇒ x = 4

AX = 1x = 4 and  XB = 2x = 2 × 4 = 8

Similarly,

AY = 1y and YC = 2y

AY = 8        …(given)

⇒ 8 = y

∴ YC = 2y = 2 × 8 = 16

∴ AC = AY + YC

AC = 8 + 16

AC = 24 cm

∆ABC is a right angled triangle.   ...(Given)

∴ By Pythagoras Theorem, we get

⇒ AB² + BC² = AC²

⇒ BC² = AC² - AB²

⇒ BC² = (24)² - (12)²

⇒ BC² = 576 - 144

⇒ BC² = 432

⇒ BC = ${\displaystyle \sqrt{432}}$

⇒ BC = ${\displaystyle 2\times 2\times 3\sqrt{3}}$

⇒ BC = ${\displaystyle 12\sqrt{3}\text{cm}}$