Question

Draw a pair of tangents to a circle of radius $5\mathrm{cm}$ which is inclined to each other at an angle of $60°$. Give the justification of the construction.

Answer 1

Step 1. Draw a pair of tangents with the help of the given statement.

To create a tangent, apply the following conditions:

1. Draw a circle with a radius of $5\mathrm{cm}$ and an $O$ in the center.
2. Take a point $A$ on the circle's perimeter and link it to $OA$.
3. At point $A$, draw a perpendicular to $AP$.
4. With $OA$, draw a radius OR with an angle of $120°$, i.e. $\left(120°-80°\right)$
5. At point $B$, draw a perpendicular to $BP$.
6. At point $P$, both perpendiculars now cross.
   Hence, $AP$ and $BP$ are the required tangents inclined at $60°$.

The diagram is shown below:

Hence, the diagram is shown below:

Step 2. Make a justification.

$\angle OAP=\angle OBP=90°$ (By construction)

$\angle AOB=120°$ (By construction)

In quadrilateral $OAPB$,

$\begin{array}{rcl}\angle APB+\angle OAP+\angle OBP+\angle AOB& =& 360°\\ & \Rightarrow & \angle APB+90°+90°+120°=360°\\ & \Rightarrow & \angle APB+300°=360°\\ & \Rightarrow & \angle APB=360°-300°\\ & \Rightarrow & \angle APB=60°\end{array}$

Hence, $AP$ and $BP$ are the required tangents inclined at $60°$.