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

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