55
Prove that the lengths of two tangents drawn from an external point to a circle are equal.
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Given: $PQ$ and $PR$ are tangents to a circle with centre $O$
To prove: $PQ = PR$.
Construction: Join $OP, OQ$ and $OR$.
Proof :In $\Delta OQP$ and $\Delta ORP$
$\angle OQP = \angle ORP = 90^\circ$
$OQ = OR$ (Radii)
$OP = OP$ (Common side)
$\implies \Delta OQP \cong \Delta ORP$ [By RHS congruence criterion]
$PQ = PR$ [CPCT]
To prove: $PQ = PR$.
Construction: Join $OP, OQ$ and $OR$.
Proof :In $\Delta OQP$ and $\Delta ORP$
$\angle OQP = \angle ORP = 90^\circ$
$OQ = OR$ (Radii)
$OP = OP$ (Common side)
$\implies \Delta OQP \cong \Delta ORP$ [By RHS congruence criterion]
$PQ = PR$ [CPCT]