  Question

Draw a circle of radius $$3 m$$ Take two prints $$P$$ and $$Q$$ on one its extended diameter each at a distance of $$7 cm$$ from its centre. Draw tangents  to the circle from these two points $$P$$ and $$Q$$.

Solution

Steps of Construction : Step I :  Taking a point $$O$$ as center , draw a circle of radius $$3 cm$$Step II : Take two points $$P$$ and $$Q$$ on one of its extended diameter such that $$OP = OQ = 7 cm$$Step III : Bisect $$OP$$ and $$OQ$$ and let $$M_1$$ and $$M_2$$ be the mid-points of $$OP$$ and $$OQ$$ respectively.Step IV : Draw a circle with $$M_1$$as center and $$M_1 P$$ as radius to intersect the circle at $$T_1$$ and $$T_2$$ Strep V  :  Joint $$PT_1$$ and $$PT_2$$Then $$PT_1$$ and $$PT_3$$ are the required tangents . Similarly the tangents $$QT_3$$ and $$QT_4$$ can be obtained.Justification ; On Joining $$OT_1$$ we find $$\angle PT_O = 90^0$$ as it is an angle  in the semicircle.$$\therefore PT_1 \bot OT_1$$Since $$OT_1$$ is a radius of the given circle , So $$PT_1$$ has to be the tangents to the circle.Similarly $$PT_2, QT_3$$ and $$QT_4$$ are also tangents to the circle. Mathematics

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