Q4:

In each of the following, give also the justification of the construction: Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of $60^{\circ}$.

Given: A circle with center $O$ and radius $r = 5\text{ cm}$. The angle between the two tangents is $60^{\circ}$.

To Find: Construct the pair of tangents and provide the geometric justification for the construction.

Step 1: Conceptual Analysis
Let the circle have center $O$. Let the two tangents meet at point $P$. Let the points of contact be $A$ and $B$.
In the quadrilateral $OAPB$:
- $\angle OAP = 90^{\circ}$ (Radius is perpendicular to the tangent at the point of contact).
- $\angle OBP = 90^{\circ}$ (Radius is perpendicular to the tangent at the point of contact).
- $\angle APB = 60^{\circ}$ (Given).
- The sum of angles in a quadrilateral is $360^{\circ}$.
Therefore, $\angle AOB = 360^{\circ} - (90^{\circ} + 90^{\circ} + 60^{\circ}) = 360^{\circ} - 240^{\circ} = 120^{\circ}$.

Step 2: Construction Procedure
1. Draw a circle of radius $5\text{ cm}$ with center $O$.
2. Draw any radius $OA$.
3. Construct an angle of $120^{\circ}$ at the center $O$ such that $\angle AOB = 120^{\circ}$.
4. At point $A$, construct a perpendicular to $OA$.
5. At point $B$, construct a perpendicular to $OB$.
6. Let these two perpendiculars intersect at point $P$. $PA$ and $PB$ are the required tangents.

Step 3: Justification
In quadrilateral $OAPB$:
- $\angle OAP = 90^{\circ}$ [By construction, as the tangent is perpendicular to the radius].
- $\angle OBP = 90^{\circ}$ [By construction, as the tangent is perpendicular to the radius].
- $\angle AOB = 120^{\circ}$ [By construction].
- The sum of interior angles of a quadrilateral is $360^{\circ}$.
- $\angle APB = 360^{\circ} - (\angle OAP + \angle OBP + \angle AOB)$
- $\angle APB = 360^{\circ} - (90^{\circ} + 90^{\circ} + 120^{\circ})$
- $\angle APB = 360^{\circ} - 300^{\circ} = 60^{\circ}$.
Thus, the tangents are inclined to each other at $60^{\circ}$.

Final Answer: The construction is justified as the angle between the tangents is calculated to be $60^{\circ}$ based on the properties of the quadrilateral $OAPB$ formed by the radii and the tangents.