Advertisements
Advertisements
प्रश्न
Prove that “The lengths of the two tangent segments to a circle drawn from an external point are equal.”
उत्तर १
Given: A circle with centre O and an external point P are given.
AP and BP are the two tangents drawn from an external point P.
To prove: AP = BP
Construction: Draw seg OA, seg OB and seg OP.
Proof: In ΔOBP and ΔOAP,
OA = OB … (Radii of the same circle)
OP = OP … (Side common to both the triangles)
∠OAP = ∠OBP = 90° … (tangent is perpendicular to the radius at the point of contact)
ΔOBP ≅ ΔOAP … (By R.H.S)
∴ AP = BP … (corresponding sides of congruent triangles)
Thus, the lengths of two tangent segments to a circle drawn from an external point are equal.
उत्तर २
Given: O is the centre of the circle and P is a point in the exterior of the circle. A and B are the points of contact of the two tangents from P to the circle.
To Prove: PA = PB
Construction: Draw seg OA, seg OB and seg OP.
Proof: Line AP ⊥ radius OA and line BP ⊥ radius OB ... (Tangent perpendicular to radius)
∴ `anglePAO = anglePBO = 90^@`
In right-angled triangles `triangle OAP` and `triangleOBP`
hypotenuse OP ≅ hypotenuse OP ...(Common side)
seg OA ≅ seg OB ...(Radii of the same circle)
`:.triangleOAP ≅ triangleOBP` ...(Hypotenuse-side of theorem)
∴ seg PA ≅ seg PB ...(c.s.c.t.)
∴ PA = PB
APPEARS IN
संबंधित प्रश्न
In Fig.3, from an external point P, two tangents PT and PS are drawn to a circle with centre O and radius r. If OP = 2r, show that ∠ OTS = ∠ OST = 30°.
If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that ∠QPR = 120°, prove that 2PQ = PO.
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
In the figure, AB and CD are common tangents to two circles of unequal radii. Prove that AB = CD.
In the given figure, AD is a diameter. O is the centre of the circle. AD is parallel to BC and ∠CBD = 32°.
Find:
- ∠OBD
- ∠AOB
- ∠BED
A right circular cone is divided into three parts by trisecting its height by two planes drawn parallel to the base. Show that the volumes of the three portions starting from the top are in the ratio 1 : 7 : 19 ?
In the given circle with center o, ∠ABC=100°, ∠ACD=40° and CT is tangent to the circle at C. find ∠ADC and ∠DCT.
In the following figure, Q is the centre of a circle and PM, PN are tangent segments to the circle. If ∠MPN = 60°, find ∠MQN.
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.
If two tangents inclined at an angle of 60° are drawn to a circle of radius 3 cm the length of each tangent is equal to ______
The length of tangents drawn from an external point to the circle ______
The number of tangents drawn at a point of the circle is/are ______
The length of a tangent drawn from a point at a distance of 10 cm of the circle is 8 cm. The radius of the circle is ______
Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle.
In a circle of radius 17 cm, two parallel chords are drawn on opposite sides of a diameter. The distance between the chords is 23 cm. If the length of one chord is 16 cm, then the length of the other is ______
If from an external point B of a circle with centre O, two tangents BC and BD are drawn such that ∠DBC = 120°, prove that BC + BD = BO, i.e., BO = 2BC.
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 ______.
Two concentric circles with centre O are of radii 3 cm and 5 cm. Find the length of chord AB of the larger circle which touches the smaller circle at P.
