Question

If \(\dfrac{AO}{OC} = \dfrac{BO}{OD} = \dfrac{1}{2}\) and \(AB = 5 \, \text{cm}\) in the following figure, then find the value of \(DC\). 

Hint:

When two sides of a quadrilateral are divided proportionally and the joining line segments are equal, the opposite sides are parallel and proportional (Thales’ Theorem).

Answer 1

Step 1: Given information.
It is given that, \[ \frac{AO}{OC} = \frac{1}{2} \quad \text{and} \quad \frac{BO}{OD} = \frac{1}{2} \] Also, \(AB = 5 \, \text{cm}\).
Step 2: Apply the concept of similar triangles.
Since \(AO/OC = BO/OD\), by the Converse of Basic Proportionality Theorem (Thales’ theorem), we have: \[ AB \parallel CD \]
Step 3: Use the property of parallel sides.
Since \(AB \parallel CD\) and \(AO/OC = 1/2\), the ratio of similarity between the two triangles \(AOB\) and \(COD\) is \[ \frac{\text{AB}}{\text{CD}} = \frac{AO}{OC} = \frac{1}{2} \]
Step 4: Find \(DC\).
\[ \frac{AB}{DC} = \frac{1}{2} \Rightarrow DC = 2 \times AB = 2 \times 5 = 10 \, \text{cm} \]
Step 5: Conclusion.
Hence, the length of \(DC\) is \(\boxed{10 \, \text{cm}}\).