Tangents

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A tangent to a circle is a line that touches the circle at exactly one point. When we have a chord through points A and X on a circle, and we move point X closer and closer to point A along the circle…

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Property: The tangent at a point to a circle is perpendicular to the radius through that point. Proof: Consider chord AX where X moves closer to A. In triangle OAX (O is center), OA = OX (radii), so …

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Step 1: Mark the given point on the circle Step 2: Draw the radius from the center to this point Step 3: At the given point, construct a perpendicular to this radius Step 4: This perpendicular line is…

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The tangents drawn at the ends of a diameter are parallel to each other. Proof: Let AB be a diameter. At A, tangent is perpendicular to radius OA. At B, tangent is perpendicular to radius OB. Since O…

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The angle at the center between two radii and the angle between the tangents at the ends of these radii are supplementary (add up to 180°). In quadrilateral OACB where tangents meet at C: - Angles at…

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Exactly two tangents can be drawn from an external point to a circle. Construction Method: Step 1: Join the external point P to the center O Step 2: Draw a circle with PO as diameter Step 3: This cir…

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Given: PA and PB are tangents from external point P to circle with center O. To prove: PA = PB Proof: In triangles OAP and OBP: - OA = OB (radii) - OP = OP (common side) - ∠OAP = ∠OBP = 90° (tangent …

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Given: Radius = 4 cm, OP = 5 cm To find: Length of tangent PT Step 1: Draw a diagram showing the right triangle OTP Step 2: Apply Pythagoras theorem in right triangle OTP: OP² = OT² + PT² Step 3: Su…