Construct a circle of radius 5 cm . Then construct a pair of tangents to the circle such that the angle between them is $$ 60^{\circ} $$

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To construct a circle of radius 5 cm and then create a pair of tangent lines to the circle that form a $$60^\circ$$ angle between them, follow these step-by-step instructions using a compass and straightedge:

### **Materials Needed**
- Compass
- Ruler or straightedge
- Protractor (optional, but helpful for verifying angles)
- Pencil
- Paper

### **Step 1: Draw the Circle**
1. **Draw the Circle with Center $$ O $$**
   - Set your compass to a radius of **5 cm**.
   - Draw a circle and label its center as point $$ O $$.

![Step 1](https://i.imgur.com/1VhO6hc.png)

### **Step 2: Choose the First Point of Tangency**
2. **Select Point $$ A $$ on the Circumference**
   - Choose any point on the circumference and label it as point $$ A $$.

![Step 2](https://i.imgur.com/mJmzuKr.png)

### **Step 3: Construct the First Tangent Line**
3. **Draw Tangent Line $$ l_1 $$ at Point $$ A $$**
   - **Construct a Perpendicular to $$ OA $$ at $$ A $$:**
     - Using a compass, draw an arc above and below point $$ A $$ intersecting $$ OA $$.
     - Without changing the compass width, draw arcs from these intersection points to create a perpendicular line.
   - Draw the tangent line $$ l_1 $$ perpendicular to $$ OA $$ at $$ A $$.

![Step 3](https://i.imgur.com/E8EHLpc.png)

### **Step 4: Determine the Second Point of Tangency**
4. **Construct a $$120^\circ$$ Angle at Center $$ O $$**
   - Since the angle between the two tangent lines needs to be $$60^\circ$$, the central angle between the two radii to the points of tangency will be $$120^\circ$$ (because the tangent lines are each perpendicular to their respective radii).
   - **Using a Protractor:**
     - Place the protractor at point $$ O $$.
     - Measure a $$120^\circ$$ angle from $$ OA $$ and mark this angle.
   - **Alternatively, Using Compass and Straightedge:**
     - Construct an equilateral triangle to achieve a $$60^\circ$$ angle and then double it to get $$120^\circ$$.

![Step 4](https://i.imgur.com/E4c2qq1.png)

5. **Mark Point $$ B $$ on the Circumference**
   - From point $$ O $$, mark point $$ B $$ on the circumference such that angle $$ AOB = 120^\circ $$.

![Step 5](https://i.imgur.com/qh7LzEC.png)

### **Step 5: Construct the Second Tangent Line**
6. **Draw Tangent Line $$ l_2 $$ at Point $$ B $$**
   - **Construct a Perpendicular to $$ OB $$ at $$ B $$:**
     - Repeat the perpendicular construction process used for point $$ A $$ to find the direction of the tangent line at $$ B $$.
   - Draw the tangent line $$ l_2 $$ perpendicular to $$ OB $$ at $$ B $$.

![Step 6](https://i.imgur.com/bBm6LiG.png)

### **Step 6: Verify the Angle Between the Tangents**
7. **Check the $$60^\circ$$ Angle Between $$ l_1 $$ and $$ l_2 $$**
   - Using a protractor, measure the angle between tangent lines $$ l_1 $$ and $$ l_2 $$ to ensure it is $$60^\circ$$.
   - This verification confirms the accuracy of your construction.

![Step 7](https://i.imgur.com/YpZBluW.png)

### **Final Result**
You have successfully constructed a circle with a radius of 5 cm and two tangent lines that intersect at a $$60^\circ$$ angle.

![Final Construction](https://i.imgur.com/u8dCjRS.png)

### **Summary of Steps**
1. Draw a circle with center $$ O $$ and radius 5 cm.
2. Select a point $$ A $$ on the circle and draw the tangent line $$ l_1 $$ at $$ A $$.
3. Construct a $$120^\circ$$ angle at $$ O $$ to locate point $$ B $$ on the circumference.
4. Draw the tangent line $$ l_2 $$ at $$ B $$.
5. Verify that the angle between $$ l_1 $$ and $$ l_2 $$ is $$60^\circ$$.

By following these steps, you can accurately create two tangent lines to a circle that form a specific angle between them.
To construct a circle with a radius of 5 cm and two tangent lines that form a $$60^\circ$$ angle between them, follow these steps: 1. **Draw the Circle**: Use a compass to draw a circle with center $$ O $$ and a radius of 5 cm. 2. **Draw the First Tangent**: At any point $$ A $$ on the circle, draw a tangent line $$ l_1 $$ perpendicular to the radius $$ OA $$. 3. **Determine the Second Point**: From center $$ O $$, mark point $$ B $$ on the circle such that the angle $$ AOB $$ is $$120^\circ$$. 4. **Draw the Second Tangent**: At point $$ B $$, draw a tangent line $$ l_2 $$ perpendicular to the radius $$ OB $$. 5. **Verify the Angle**: Ensure that the angle between $$ l_1 $$ and $$ l_2 $$ is $$60^\circ$$. This construction results in two tangent lines to the circle that intersect at a $$60^\circ$$ angle.

Construct a circle of radius 5 cm . Then construct a pair of tangents to the circle such that the angle between them is \( 60^{\circ} \)

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