Question

In the given figure, \(OA \times OB = OC \times OD\). Which of the following option is correct ?

• A. A = C
• B. A = B
• C. A = D
• D. OAD OBC

Hint:

When dealing with similarity, the order of the letters is extremely important.
From \( \frac{OA}{OC} = \frac{OD}{OB} \), point \( A \) corresponds to \( C \), and point \( D \) corresponds to \( B \).
This immediately implies that \( \angle A = \angle C \) and \( \angle D = \angle B \).
Always write out the ratio first to avoid confusion with vertex matching.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The topic of this question is Triangles (Similarity of Triangles).
We are given two intersecting lines \( AB \) and \( CD \) at point \( O \), forming a geometric configuration.
We are given the algebraic relation \( OA \times OB = OC \times OD \).
We need to determine the correct relationship between the angles of the triangles formed.

Step 2: Key Formula or Approach:
- Rearrange the given multiplication relation into ratio form to identify proportional sides.
- Use vertically opposite angles to find equal angles:
\[ \angle AOD = \angle COB \]
- Apply the Side-Angle-Side (SAS) similarity criterion to establish similarity between the triangles.

Step 3: Detailed Explanation:
1. We are given the relation:
\[ OA \times OB = OC \times OD \]
2. Rearrange this relation by dividing both sides by \( OC \times OB \):
\[ \frac{OA}{OC} = \frac{OD}{OB} \]
3. Now, consider the two triangles \( \Delta OAD \) and \( \Delta OCB \):
- From our rearranged equation, the ratios of the corresponding sides are equal:
\[ \frac{OA}{OC} = \frac{OD}{OB} \]
- The angle included between these sides is vertically opposite and therefore equal:
\[ \angle AOD = \angle COB \]
4. By the Side-Angle-Side (SAS) similarity criterion, we can conclude:
\[ \Delta OAD \sim \Delta OCB \]
5. Since corresponding angles of similar triangles are equal, we can write:
- \( \angle OAD = \angle OCB \implies \angle A = \angle C \)
- \( \angle ODA = \angle OBC \implies \angle D = \angle B \)
6. Let's analyze the given options:
- Option (A) states \(\angle A = \angle C\). This is correct according to our similarity.
- Option (B) states \(\angle A = \angle B\). This is incorrect.
- Option (C) states \(\angle A = \angle D\). This is incorrect.
- Option (D) states \(\Delta OAD \sim \Delta OBC\). The correct similarity correspondence is \(\Delta OAD \sim \Delta OCB\). Since the vertices are not in the correct corresponding order, Option (D) is mathematically incorrect.

Step 4: Final Answer:
The correct option is (A), which is \(\angle A = \angle C\).