Theorem - The Length of Two Tangent Segments Drawn from a Point Outside the Circle Are Equal
A circle can have atmost two parallel tangents.
l and m are two tangents, radii OA and OB are ⊥ m and l respectively. As studied in 9th class, if the sum of co-interior angles between two lines is `180°` then the lines are parallel.
The point at which tangent touches a circle is known as point of contact. Here, point A and B are points of contact.
Only two tangents can be drawn from any external point of a circle.
The length of the segment of the tangent from the external point P and the point of contact with the circle is called the length of the tangent from the point P to the circle.
The lengths PT1 and PT2 are equal, this we will learn in the further concepts.
Theorem- The lengths of tangents drawn from an external point to a circle are equal.
Given: A circle C (o,r) and an external point P. PQ and PR are lengths of circle.
To prove: PQ=PR
Proof: In ΔPOQ and ΔPOR
∠PRO and ∠PQO are right angled triangles (Theorem 1)
OQ = OR (Radii of the same circle)
Therefore, ∆ OQP ≅ ∆ ORP (Two right triangles are congruent if the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and the corresponding side of the other triangle.)
This gives PQ = PR (CPCT)
Shaalaa.com | Lengths of two tangents drawn from external point to a circle are SAME
In the following figure, PQ = QR, `∠`RQP = 68° , PC and CQ are tangents to the circle with centre O
M and N are the midpoints of chords AB and CD . The line MN passes through the centre O . Prove that AB || CD.
In the given figure, PQ is a tangent to the circle at A. AB and AD are bisectors of ∠CAQ and ∠PAC. If ∠BAQ = 30°, prove that : BD is diameter of the circle.
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In the figure given below, an AD is a diameter. O is the centre of the circle AD is parallel to BC and `angle CBD` = 32°. Find `angle OBD`