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source: ncert
topic: geometry
class_level: class_10
chapter: ch10
difficulty: intermediate
---
Class 10 | Ch10: Circles
KEY DEFINITIONS:
Tangent: line touching circle at exactly ONE point (point of tangency/contact).
Secant: line intersecting circle at TWO points.
A tangent is perpendicular to radius at point of contact.
THEOREMS:
Theorem 1: Tangent at any point of circle is perpendicular to radius through that point.
Proof: Any point on tangent other than contact point is outside circle (farther from centre).
So perpendicular from centre to tangent = shortest distance = radius. Hence radius ⊥ tangent.
Theorem 2: Lengths of tangents drawn from external point are EQUAL.
If PA and PB are tangents from P to circle with centre O:
In triangles OAP and OBP: OA=OB (radii), OP=OP (common), angle OAP=angle OBP=90°.
By RHS: triangles congruent → PA=PB (CPCT).
Also: OP bisects angle APB and angle AOB.
LENGTH OF TANGENT FROM EXTERNAL POINT:
If d=distance from external point to centre, r=radius:
Length of tangent=sqrt(d^2-r^2).
ANGLE IN ALTERNATE SEGMENT (Tangent-Chord angle):
Angle between tangent and chord = inscribed angle in alternate segment.
SOLVED EXAMPLES:
Example 1: Point P is 17cm from centre. Radius=8cm. Find tangent length.
Tangent=sqrt(17^2-8^2)=sqrt(289-64)=sqrt(225)=15cm.
Example 2: Two tangents PA=PB from external point P. PA=12cm, OA=5cm. Find OP.
OP^2=OA^2+PA^2=25+144=169. OP=13cm.
Example 3: Tangents from P touch circle at A and B. Angle APB=60°. Find angle AOB.
Since PA=PB, triangle PAB is isosceles. Angle OAP=90°.
In quadrilateral OAPB: angle AOB+angle OBP+angle APB+angle OAP=360°.
angle AOB+90°+60°+90°=360° → angle AOB=120°.
COMMON MISTAKES:
Tangent from EXTERNAL point has equal length — this fails for internal points.
Tangent ⊥ radius at POINT OF CONTACT (not at every point of tangent).
Length of tangent uses Pythagoras: sqrt(d^2-r^2), not d-r. |