Question

In the given figure, ABC is an equilateral triangle of side length 30 cm. XY is parallel to BC, XP is parallel to AC and YQ is parallel to AB. If (XY + XP + YQ) is 40 cm, then what is PQ equal to? 

(a) 5 cm 

(b) 12 cm 

(c) 15 cm 

(d) None of these

Answer 1

(d) None of these

AB || YQ ⇒ ∠XBP = ∠YQC 

(corresponding angles) and 

XP || AC ⇒ ∠XPB = ∠YCQ 

ΔABC being an equilateral triangle, 

∠B = ∠C = 60º 

⇒ ∠XPB = ∠YQC = 60° 

⇒ ∠XPB = ∠QYC = 60º 

⇒ ΔXBP and ΔYQC are equilateral triangles. 

Now, XY || BC ⇒ \(\frac{AX}{AB}=\frac{XY}{BC}\) ⇒ AX = XY            (∵ AB = BC) 

Also, XY + XP + YQ = 40 ⇒ AX + XB + YQ = 40     (∵ AX = XY, XP = XB) 

⇒ AB + YQ = 40 ⇒ YQ = 40 – AB = 40 – 30 = 10 

∴ XP = YQ = 10 cm. 

⇒ BP = QC = 10 cm (ΔXBP and ΔYQC are equilateral) 

⇒ PQ = BC – (BP + QC) = 30 – 10 – 10 = 10 cm.