Question

In the given figure, \(\Delta ODC \sim \Delta OBA\). If \(\angle BOC = 110^\circ\), \(\angle ODC = 45^\circ\) and \(AB = 2 CD\), then find (i) \(m\angle OAB\) (ii) \(OB : OD\).

Hint:

Always write out the similarity statement carefully.
The order of letters in \(\Delta ODC \sim \Delta OBA\) tells you exactly which vertices correspond to each other:
\(O \leftrightarrow O\), \(D \leftrightarrow B\), and \(C \leftrightarrow A\).
This makes finding corresponding angles and side ratios highly systematic.

Answer 1

Correct Answer: 1

Step 1: Understand the Given Information.
We are given $\Delta ODC \sim \Delta OBA$, $\angle BOC = 110^\circ$, $\angle ODC = 45^\circ$, and $AB = 2CD$. We need to find (i) $\angle OAB$ and (ii) $OB : OD$.
Step 2: Find $\angle DOC$ Using Angles at O.
Since $\angle BOC = 110^\circ$, and $B$, $O$, $D$ are positioned such that $\angle DOC = 180^\circ - \angle BOC$ (angles on a straight line or supplementary): \[ \angle DOC = 180^\circ - 110^\circ = 70^\circ \]
Step 3: Find $\angle DCO$ in Triangle ODC.
In $\Delta ODC$: \[ \angle ODC + \angle DCO + \angle DOC = 180^\circ \] \[ 45^\circ + \angle DCO + 70^\circ = 180^\circ \] \[ \angle DCO = 180^\circ - 115^\circ = 65^\circ \]
Step 4: Use Similarity to Find $\angle OAB$.
Since $\Delta ODC \sim \Delta OBA$, corresponding angles are equal: \[ \angle ODC \leftrightarrow \angle OBA, \quad \angle DCO \leftrightarrow \angle OAB, \quad \angle DOC \leftrightarrow \angle BOA \] Therefore: \[ \angle OAB = \angle DCO = 65^\circ \]
Step 5: Find the Ratio $OB : OD$.
Since $\Delta ODC \sim \Delta OBA$, the sides are proportional. Corresponding sides: $OD \leftrightarrow OB$ and $DC \leftrightarrow BA$. Given $AB = 2CD$: \[ \frac{OB}{OD} = \frac{BA}{DC} = \frac{AB}{CD} = \frac{2CD}{CD} = 2 \] So $OB : OD = 2 : 1$.
Step 6: State the Final Answers.
(i) $\angle OAB = 65^\circ$ and (ii) $OB : OD = 2 : 1$. \[ \boxed{\angle OAB = 65^\circ \text{ and } OB : OD = 2 : 1} \]