In a circle, a chord is drawn whose length is equal to the radius of the circle. A tangent is drawn at one end of this chord. What is the measure of the angle between the tangent and the chord?

Introduction / Context:
This circle geometry problem involves a chord whose length equals the radius and a tangent drawn at one end of that chord. It tests knowledge of the tangent chord theorem, which connects the angle between a tangent and a chord with an angle inside the circle in the alternate segment.

Given Data / Assumptions:

• A circle has radius r.
• A chord AB is drawn such that AB = r.
• A tangent is drawn at point A, one end of the chord AB.
• We must find the angle between this tangent and the chord AB.

Concept / Approach:
First, note that OA and OB are radii of the circle (O is the centre), and AB is a chord equal in length to the radius. Thus triangle AOB has all three sides equal to r, making it an equilateral triangle, so each angle in triangle AOB is 60°. The tangent at point A is perpendicular to radius OA, forming a 90° angle between OA and the tangent. The angle between the tangent at A and chord AB equals the difference between 90° and angle OAB in triangle AOB. Because triangle AOB is equilateral, angle OAB is 60°.

Step-by-Step Solution:
Step 1: Identify triangle AOB where OA and OB are radii and AB is a chord equal to the radius, so OA = OB = AB = r. Step 2: Conclude triangle AOB is equilateral, hence each internal angle is 60°. Step 3: The radius OA is perpendicular to the tangent at A, so the angle between OA and the tangent is 90°. Step 4: At point A, angle between the tangent and chord AB equals 90° minus angle OAB because OAB is inside the triangle between OA and AB. Step 5: Angle OAB = 60°, so required angle = 90° − 60° = 30°.

Verification / Alternative Check:
The tangent chord theorem states that the angle between a tangent and a chord through the point of contact equals the angle in the alternate segment of the circle that subtends the same chord. Here, the angle between the tangent at A and chord AB equals the angle in the circle at point C on the circumference such that chord AB subtends angle ACB. In equilateral triangle AOB, the corresponding inscribed angle at C is 30°, confirming the computed value.

Why Other Options Are Wrong:
45°, 60°, and 75° do not arise from the perpendicular tangent and 60° vertex angle inside the equilateral triangle. In particular, 60° would incorrectly assume that the angle between tangent and chord equals the internal angle at the centre or at the vertex of the equilateral triangle, which is not true in this configuration.

Common Pitfalls:
Many learners forget that the tangent is perpendicular to the radius and either use the 60° angle directly or misapply the tangent chord theorem. Another frequent mistake is confusing the central angle AOB (60°) with the angle between the tangent and the chord. Always remember that the radius to the point of tangency forms a right angle with the tangent, and work from there.

Final Answer:
The angle between the tangent and the chord is 30°.