Question

Construct a pair of tangents to a circle of radius $4cm$ such that the acute angle between the tangents is ${45}^{\circ }$.

Answer 1

Step 1: Place the compass on a point $O$ and draw a circle with radius $=4cm$.

Step 2: Take any point $A$ on the circle. Draw ray $OA$.

Step 3: Place the compass on point $A$ and draw two equidistant arcs on line $OA$ to intersect $OA$ in points $D$ and $E$.

Step 4: From point $D$, mark two arcs above and below line $OA$. Similarly, mark two arcs from point $E$ such that they intersect the arcs marked from point $D$.

Step 5: Join the intersection points of the arcs to obtain one tangent from $A$.

Step 6: Draw ray $OB$ such that $\angle AOB=(180-45{)}^{\circ }={135}^{\circ }$.

Step 7: Repeat steps $3,4$ and $5$ for point $B$ in place of point $A$ to obtain the tangent at point $B$

Step 8: Let the tangent from point $B$ intersect the tangent from point $A$ at point $P$.

$PA$ and $PB$ are the required tangent. The angle between $PA$ and $PB$ is ${45}^{\circ }$.