📌 Core Concept
The problem involves similar triangles formed by the sides of a trapezium. When the non-parallel sides of a trapezium are extended, they form similar triangles with the bases of the trapezium. The ratio of similarity is equal to the ratio of the lengths of the parallel sides.
🔢 Step-by-Step Solution
1
Identify Similar Triangles:
Triangles
△EAB and
△ECD are similar because
AB∥CD.
Ratio of similarity
k=CDAB=12=2.
2
Express Sides in Terms of Ratio:
Let
EA=x and
EB=y.
Since
△EAB is similar to
△ECD with ratio 2, we have:
EDEA=2andECEB=2
Let
AD=m and
BC=n. Then:
ED=EA−AD=x−mandEC=EB−BC=y−n
From the similarity ratio:
x=2mandy=2n
3
Use Perimeter Information:
Perimeter of trapezium
ABCD:
AB+BC+CD+DA=2+n+1+m=6⟹m+n=3
4
Calculate Perimeter of
△AEB:
Perimeter
P of
△AEB:
P=EA+EB+AB=2m+2n+2=2(m+n)+2=2(3)+2=8 cm ⚡ 30-Second Shortcut
Since the ratio of similarity is 2, the perimeter of
△AEB is twice the sum of the non-parallel sides of the trapezium plus the length of
AB. Given the perimeter of the trapezium is 6, the non-parallel sides sum to 3, making the perimeter of
△AEB2×3+2=8.
🎯 Final Answer
Correct Answer: A
