Question

In trapezium ABCD, M and N are mid-points of AC and BD, and AB || CD. Find

1. MN if AB = 5.5 cm, CD = 7.5 cm
2. AB if CD = 16 cm, MN = 10 cm
3. CD if AB = 14 cm, MN = 16.5 cm

Answer 1

We are given a trapezium ABCD with AB || CD and M and N are the midpoints of diagonals AC and BD, respectively. We need to find the length of MN, which is the segment joining the midpoints of the diagonals.

There is a well-known property for the length of the line joining the midpoints of the diagonals in a trapezium:

`MN = (AB + CD)/2`

Case i. AB = 5.5 cm, CD = 7.5 cm

Using the formula for MN:

`MN = (AB + CD)/2`

= `(5.5 + 7.5)/2`

= `13/2` 

= 6.5 cm

Case ii. AB = 16 cm, MN = 10 cm

Rearranging the formula to find CD:

`MN = (AB + CD)/2`

`10 = (16 + CD)/2`

Multiplying both sides by 2:

20 = 16 + CD

Solving for CD:

CD = 20 – 16 = 4 cm

Case iii. AB = 14 cm, MN = 16.5 cm

Again, using the formula for MN and solving for CD:

`MN = (AB + CD)/2`

`16.5 = (14 + CD)/2`

Multiplying both sides by 2:

33 = 14 + CD

Solving for CD:

CD = 33 – 14 = 19 cm