JEE Advanced Maths: Geometry (Trapezium) questions with solutions

Q1. In a trapezium ABCD with AB parallel to DC, E and F are the midpoints of the two diagonals. Given DC = 60 and EF = 5, find the length of AB.

1. 40
2. 45
3. 50
4. 55

Answer: 50

In a trapezium, the line segment joining the midpoints of the diagonals is parallel to the two bases and its length equals half the difference of the parallel sides: EF = |DC - AB|/2. So 5 = (60 - AB)/2, giving 60 - AB = 10, hence AB = 50.

Q2. In trapezium ABCD, AB is parallel to CD. Given that angle ADC = 2 * angle ABC, with AD = a cm and CD = b cm, find the length of AB in cm.

1. a/2 + 2b
2. a + b
3. (2/3)*a + b
4. a + (2/3)*b

Answer: a + b

Because AB || CD, co-interior and alternate angle relations combine with angle ADC = 2*angle ABC to make a sub-triangle isosceles, so AB splits into a segment equal to CD (b) plus a segment equal to AD (a).