A circle is the set of all points in a plane that are equidistant from a fixed point called the centre. The constant distance is called the radius. In this chapter, we study tangents and their properties in detail.
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Key Concepts and Definitions
Chord: A line segment joining two points on a circle.
Diameter: The longest chord, passing through the centre. Diameter = 2 × radius.
Arc: A part of the circle's circumference.
Secant: A line that intersects the circle at two distinct points.
Tangent: A line that touches the circle at exactly one point. That point is called the point of tangency.
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Theorems on Tangents
- Theorem 1 (Key): The tangent at any point on a circle is perpendicular to the radius drawn to the point of tangency.
- If PA is a tangent at A, then OA is perpendicular to PA, where O is the centre.
Theorem 2: A line drawn through the end-point of a radius, perpendicular to it, is a tangent to the circle.
- Theorem 3 (Equal Tangents): The lengths of two tangents drawn from an external point to a circle are equal.
- If PA and PB are tangents from external point P, then PA = PB.
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Key Results
- The angle between a tangent to a circle and the chord drawn through the point of tangency equals the angle in the alternate segment. (Tangent-Chord Angle)
- If O is the centre and P is external point with tangents PA and PB, then OP bisects angle APB and angle AOB.
- PA2 = PO2 - OA2 (by Pythagoras, since angle OAP = 90°)
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Worked Examples
A tangent PQ is drawn to a circle with centre O at point Q. If OQ = 5 cm and OP = 13 cm, find PQ.
- Angle OQP = 90° (radius perpendicular to tangent)
- PQ = √(OP2 - OQ2) = √(169 - 25) = √(144) = 12 cm
Two tangents PA and PB are drawn from an external point P to a circle of radius 4 cm. If PA = 3 cm, find the distance from P to the centre.
- OP2 = OA2 + PA2 = 16 + 9 = 25 => OP = 5 cm
From an external point P, two tangents PA and PB are drawn to a circle. Show that PA = PB.
- In triangles OAP and OBP: OA = OB (radii), OP = OP (common), angle OAP = angle OBP = 90°.
- By RHS congruence, triangles OAP and OBP are congruent => PA = PB.
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = BC + AD.
- Since tangents from an external point are equal:
- From A: AP = AS; From B: BP = BQ; From C: CR = CQ; From D: DR = DS
- AB + CD = AP + PB + CR + RD = AS + BQ + CQ + DS = (AS + DS) + (BQ + CQ) = AD + BC
The length of the tangent from a point P to a circle of radius 5 cm is 12 cm. Find the distance of P from the centre.
- OP = √(OA2 + PA2) = √(25 + 144) = √(169) = 13 cm
Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle that is a tangent to the smaller circle.
- The perpendicular from centre to chord = 3 cm (radius of smaller circle)
- Half-chord = √(52 - 32) = √(16) = 4 cm
- Full chord = 8 cm
PA and PB are tangents from an external point P. If angle APB = 60°, find angle PAB.
- Triangle PAB is isosceles (PA = PB), angle APB = 60°.
- Angles PAB = PBA = (180° - 60°)/2 = 60°. So triangle PAB is equilateral!
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Common mistakes
> Common mistakes: Students forget that the radius to the point of tangency is always perpendicular to the tangent. Never assume two tangents from the same external point are equal without noting that they must originate from the same external point. Also, in problems involving quadrilaterals circumscribing circles, label tangent lengths carefully.
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Summary
A tangent touches a circle at exactly one point. The radius to the point of contact is perpendicular to the tangent. Tangents from an external point are equal in length. These two key results are used to solve most problems in this chapter. The Pythagoras theorem is frequently applied since tangent and radius form a right angle.