Draw a pair of tangents to a circle which are inclined to each other at an angle of 30°. What should be the angle between two radii?

Answer

To solve the problem of drawing a pair of tangents to a circle that are inclined to each other at an angle of 30°, we need to determine the angle between the two radii drawn to the points of contact of the tangents. Here’s a step-by-step solution:

### Step 1: Understand the Geometry
We have a circle and we want to draw two tangents that meet at an angle of 30°. The points where the tangents touch the circle will be denoted as points A and B.

**Hint:** Remember that the radius drawn to the point of tangency is perpendicular to the tangent line.

### Step 2: Identify Angles in the Quadrilateral
When we draw the tangents at points A and B, we can form a quadrilateral AOBT, where O is the center of the circle, and T is the point where the tangents meet. The angles at points A and B (the angles between the radius and the tangent) are both 90°.

**Hint:** Use the property of tangents that states the radius to the point of tangency is perpendicular to the tangent.

### Step 3: Set Up the Equation
The sum of the angles in quadrilateral AOBT is 360°. We know:
- Angle A = 90° (radius to tangent at A)
- Angle B = 90° (radius to tangent at B)
- Angle T = 30° (the angle between the two tangents)

Let angle AOB be denoted as x. We can set up the equation:
\[ 90° + x + 90° + 30° = 360° \]

**Hint:** Remember to include all angles when calculating the total sum of angles in a quadrilateral.

### Step 4: Solve for x
Now, simplifying the equation:
\[ 180° + x + 30° = 360° \]
\[ x + 210° = 360° \]
\[ x = 360° - 210° \]
\[ x = 150° \]

**Hint:** Isolate x to find the angle between the two radii.

### Step 5: Conclusion
The angle between the two radii OA and OB, which meet at the center O and extend to points A and B where the tangents touch the circle, is 150°.

**Final Answer:** The angle between the two radii is 150°.