ABCD is a trapezium in which AB is parallel to CD. The sides AD and BC, when extended, intersect at point E. If AB = 3 cm, CD = 2 cm and the perimeter of ABCD is 10 cm, then the perimeter, in cm, of triangle AEB is:
Answer and solution
Correct answer: 18
Solution
Answer: 18
1. Since AB is parallel to CD, triangles EAB and EDC are similar.
2. Their corresponding bases are 3 cm and 2 cm. So their perimeters are also in the ratio 3 : 2.
3. The two slant sides of the trapezium together measure:
4. 10 − 3 − 2 = 5 cm.
5. Let the perimeter of the larger triangle EAB be P. The smaller triangle's perimeter is 2P/3.
6. The difference between these two perimeters is 5 + (3 − 2) = 6 cm.
7. So,
8. P − 2P/3 = 6
P/3 = 6
P = 18 cm.
9. Answer: 18 (TITA).
Why this method works:
Parallel bases create similar triangles with vertex E. Because AB is the longer base, triangle EAB is larger and ECD is smaller. The trapezium’s slant sides are the differences between the corresponding triangle sides.
Independent check and common trap:
AD+BC=5 and the larger triangle’s two slant sides are three times that difference, totaling 15. Add AB=3 to obtain perimeter 18. This is a second route to the same result without solving for either slant side separately.
Apply: Scale ratio 3: 2. Slant-side difference=10−3−2=5; base difference 1. Triangle perimeter difference 6 is one part →larger perimeter 3×6=18.
Check: AD+BC=5 and the larger triangle’s two slant sides are three times that difference, totaling 15. Add AB=3 to obtain perimeter 18. This is a second route to the same result without solving for either slant side separately.
1. Since AB is parallel to CD, triangles EAB and EDC are similar.
2. Their corresponding bases are 3 cm and 2 cm. So their perimeters are also in the ratio 3 : 2.
3. The two slant sides of the trapezium together measure:
4. 10 − 3 − 2 = 5 cm.
5. Let the perimeter of the larger triangle EAB be P. The smaller triangle's perimeter is 2P/3.
6. The difference between these two perimeters is 5 + (3 − 2) = 6 cm.
7. So,
8. P − 2P/3 = 6
P/3 = 6
P = 18 cm.
9. Answer: 18 (TITA).
Why this method works:
Parallel bases create similar triangles with vertex E. Because AB is the longer base, triangle EAB is larger and ECD is smaller. The trapezium’s slant sides are the differences between the corresponding triangle sides.
Independent check and common trap:
AD+BC=5 and the larger triangle’s two slant sides are three times that difference, totaling 15. Add AB=3 to obtain perimeter 18. This is a second route to the same result without solving for either slant side separately.
G Strategy shortcut
Spot: Geometry – Trapezium.Apply: Scale ratio 3: 2. Slant-side difference=10−3−2=5; base difference 1. Triangle perimeter difference 6 is one part →larger perimeter 3×6=18.
Check: AD+BC=5 and the larger triangle’s two slant sides are three times that difference, totaling 15. Add AB=3 to obtain perimeter 18. This is a second route to the same result without solving for either slant side separately.
G Strategy: Spot: Geometry – Trapezium. Apply: Scale ratio 3: 2. Slant-side difference=10−3−2=5; base difference 1. Triangle perimeter difference 6 is one part →larger perimeter 3×6=18. Check: AD+BC=5 and the larger triangle’s two slant sides are three times that difference, totaling 15. Add AB=3 to obtain perimeter 18. This is a second route to the same result without solving for either slant side separately.