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: Test the relation with a concrete valid example first.
Before doing the general proof, check the claim with actual numbers that satisfy $OA \times OB = OC \times OD$. Take $OA = 6$, $OB = 2$, $OC = 4$, $OD = 3$. Check: $OA \times OB = 6 \times 2 = 12$ and $OC \times OD = 4 \times 3 = 12$, so the condition holds for these values. This numeric case will guide which angle relation must be true in general.

Step 2: Rearrange the given relation into a ratio.
\[ OA \times OB = OC \times OD \]
Divide both sides by $OB \times OC$:
\[ \frac{OA}{OC} = \frac{OD}{OB} \]
With our numeric check: $\frac{OA}{OC} = \frac{6}{4} = 1.5$ and $\frac{OD}{OB} = \frac{3}{2} = 1.5$, confirming the ratio equality.

Step 3: Identify the angle common to both triangles formed at the intersection point.
Lines $AB$ and $CD$ intersect at $O$, so angle $AOD$ and angle $COB$ are vertically opposite angles, and vertically opposite angles are always equal:
\[ \angle AOD = \angle COB \]

Step 4: Apply SAS similarity.
In triangles $OAD$ and $OCB$, the sides around the equal angle are in the same ratio ($\frac{OA}{OC} = \frac{OD}{OB}$) and the included angle $\angle AOD = \angle COB$ is equal. By the Side-Angle-Side similarity rule:
\[ \Delta OAD \sim \Delta OCB \]

Step 5: Read off the matching angle and rule out the other options.
Since the vertices correspond as $O \leftrightarrow O$, $A \leftrightarrow C$, $D \leftrightarrow B$, we get angle $A$ = angle $C$, which is option (A).
Option (B), angle $A$ = angle $B$, would require $A$ and $B$ to be corresponding vertices, but $A$ corresponds to $C$ here, not $B$, so this is not implied by the given equality.
Option (C), angle $A$ = angle $D$, again puts $A$ against the wrong vertex, so it does not follow either.
Option (D) writes the similarity as $\Delta OAD \sim \Delta OBC$, but our derivation shows the correct vertex order is $\Delta OAD \sim \Delta OCB$ (not $OBC$), so the vertices are out of order and this statement is not correct as written.

Final Answer:
The correct relation is angle $A$ = angle $C$, so option (A) is correct.
\[ \boxed{\angle A = \angle C} \]