| --- |
| 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. |
