(D) We know that the radius of a circle is perpendicular to the tangent at the point of contact.Therefore,$OP \perp PQ$.This forms a right-angled tria

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(D) We know that the radius of a circle is perpendicular to the tangent at the point of contact.Therefore,$OP \perp PQ$.This forms a right-angled triangle $\triangle OPQ$,where $OP = 5 \, cm$ (radius) and $OQ = 12 \, cm$ (hypotenuse).By applying the Pythagoras theorem in $\triangle OPQ$:$OP^2 + PQ^2 = OQ^2$$5^2 + PQ^2 = 12^2$$25 + PQ^2 = 144$$PQ^2 = 144 - 25$$PQ^2 = 119$$PQ = \sqrt{119} \, cm$.

Class 10 Mathematics Circles Textbook - Circles

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$A$ tangent $PQ$ at a point $P$ of a circle of radius $5 \, cm$ meets a line through the centre $O$ at a point $Q$ so that $OQ = 12 \, cm$. The length of $PQ$ is (in $cm$):

A
$12$
B
$13$
C
$8.5$
D
$\sqrt{119}$

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Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

Difficult

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Fill in the blanks:
$(i)$ $A$ circle can have $......$ parallel tangents at the most.
$(ii)$ The common point of a tangent to a circle and the circle is called $......$

Medium

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Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Difficult

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If tangents $PA$ and $PB$ from a point $P$ to a circle with centre $O$ are inclined to each other at an angle of $80^{\circ}$,then $\angle POA$ is equal to ......$^{\circ}$

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Two tangents $TP$ and $TQ$ are drawn to a circle with centre $O$ from an external point $T$. Prove that $\angle PTQ = 2 \angle OPQ$.

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$A$ tangent $PQ$ at a point $P$ of a circle of radius $5 \, cm$ meets a line through the centre $O$ at a point $Q$ so that $OQ = 12 \, cm$. The length of $PQ$ is (in $cm$):

Similar Questions

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

Fill in the blanks:$(i)$ $A$ circle can have $......$ parallel tangents at the most.$(ii)$ The common point of a tangent to a circle and the circle is called $......$

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

If tangents $PA$ and $PB$ from a point $P$ to a circle with centre $O$ are inclined to each other at an angle of $80^{\circ}$,then $\angle POA$ is equal to ......$^{\circ}$

Two tangents $TP$ and $TQ$ are drawn to a circle with centre $O$ from an external point $T$. Prove that $\angle PTQ = 2 \angle OPQ$.

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Fill in the blanks:
$(i)$ $A$ circle can have $......$ parallel tangents at the most.
$(ii)$ The common point of a tangent to a circle and the circle is called $......$