Assertion (A) TA and TB are two tangents drawn from an external point T to a circle with centre 'O'. If ∠TBA=75∘ then ∠ABO=25∘.
Reason (R) The tangent drawn at any point of a circle is perpendicular to the radius through the point of contact.
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Step-by-Step Solution
Step 1: Analyze the given information
We are given that TA and TB are tangents drawn from an external point T to a circle with center O. We are also given that the angle ∠TBA=75∘. We need to verify if the assertion that ∠ABO=25∘ is true.
Step 2: Apply properties of tangents and radii
According to the reason (R), the tangent drawn at any point of a circle is perpendicular to the radius through the point of contact. Therefore, the radius OB is perpendicular to the tangent TB at the point of contact B, which means ∠OBT=90∘.
Step 3: Calculate ∠ ABO
We know that ∠OBT is the sum of ∠OBA and ∠TBA. Substituting the known values, we get 90∘=∠OBA+75∘. Solving for ∠OBA, we find that ∠OBA=15∘.
Step 4: Evaluate the assertion
Our calculation shows that ∠ABO=15∘. The assertion states that ∠ABO=25∘. Since 15∘=25∘, the assertion (A) is false.
Step 5: Evaluate the reason
The reason (R) states that the tangent drawn at any point of a circle is perpendicular to the radius through the point of contact. This is a fundamental theorem in geometry and is a true statement.