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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.
Solution
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 = `sqrt(432)`
⇒ BC = `2 xx 2 xx 3sqrt3`
⇒ BC = `bb(12sqrt3 " cm")`
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