Advertisements
Advertisements
प्रश्न
Prove that the lengths of the tangents drawn from an external point to a circle are equal.
उत्तर
Given:
TP and TQ are two tangents drawn from an external point T to the circle C (O, r).
To prove: TP = TQ
Construction: Join OT.
Proof:
We know that a tangent to the circle is perpendicular to the radius through the point of contact.
∴ ∠OPT = ∠OQT = 90°
In ΔOPT and ΔOQT,
OT = OT ...(Common)
OP = OQ ...(Radius of the circle)
∠OPT = ∠OQT ...(90°)
∴ ΔOPT ≅ ΔOQT ...(RHS congruence criterion)
⇒ TP = TQ ...(CPCT)
Hence, the lengths of the tangents drawn from an external point to a circle are equal.
APPEARS IN
संबंधित प्रश्न
In the following figure, Q is the center of the circle. PM and PN are tangents to the circle. If ∠MPN = 40° , find ∠MQN.
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
In the given figure, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to ______.
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
In the given figure, XY and X’Y’ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X’Y’ at B. Prove that ∠AOB = 90°
In the given figure PA = 6, PB = 4 and PC = 8. Find PD
In Fig. 4, a circle is inscribed in a ΔABC having sides BC = 8 cm, AB = 10 cm and AC = 12 cm. Find the lengths BL, CM and AN.
The length of the tangent from an external point P on a circle with centre O is ______
In the given figure, PA and PB are tangents from external point P to a circle with centre C and Q is any point on the circle. Then the measure of ∠AQB is ______.
PA and PB are tangents drawn to a circle of centre O from an external point P. Chord AB makes an angle of 30° with the radius at the point of contact. If length of the chord is 6 cm, find the length of the tangent PA and the length of the radius OA.
