Question

In the given figure, AB $\parallel$ DC. If OB = 3OD and CD = 1.8 cm, then find the length AB.

Hint:

Whenever you have a trapezium with parallel bases, the triangles formed by the segments of the diagonals are always similar.
This means the ratio of the bases is equal to the ratio of the diagonal segments: \(AB / CD = OB / OD\).

Answer 1

Correct Answer: 5.4

Step 1: Understand the configuration.
AB is parallel to DC. The diagonals of trapezium ABCD intersect at O. OB = 3OD and CD = 1.8 cm. We need to find AB.
Step 2: Identify similar triangles.
Since AB \(\parallel\) DC, triangles AOB and COD are similar by AA criterion (vertically opposite angles at O, and alternate interior angles with the parallel lines).
Step 3: Write the ratio of similarity.
The ratio of similarity \(= \frac{OB}{OD} = \frac{3}{1}\) (since OB = 3OD).
Step 4: Apply the ratio to find AB.
Since triangles AOB and COD are similar: \(\frac{AB}{CD} = \frac{OB}{OD} = \frac{3}{1}\).
Step 5: Substitute CD = 1.8 cm.
\(AB = 3 \times CD = 3 \times 1.8 = 5.4 \text{ cm}\).
Step 6: State the final answer.
\[ \boxed{AB = 5.4 \text{ cm}} \]