The method of judging the tangent of a straight line and a circle: if there is only one common point between the straight line and the circle, the straight line is tangent to the circle; The distance from the center of the circle to a straight line is equal to the radius of the circle, so this straight line is the tangent of the circle; The straight line perpendicular to the radius at the outer end of the radius is the tangent of the circle.
The tangent of a circle is perpendicular to the radius passing through the tangent point.
Inference 1: A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point.
Inference 2: A straight line passing through the tangent and perpendicular to the tangent must pass through the center of the circle.
Extended data:
Judgment theorem of tangent: the straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle. The tangent of a circle is perpendicular to the radius of the tangent point of the circle.
Geometric language:
∵l⊥OA, point A is on ⊙ O.
∴ The straight line L is tangent to⊙ O (tangent judgment theorem)
Tangent Theorem: The tangent of a circle is perpendicular to the radius passing through the tangent point.
Geometric language:
∵OA is the radius ⊙ O, and the straight line L cuts ⊙ O at point A.
∴l ⊥OA (tangent property theorem)
Inference 1 The diameter passing through the center of the circle and perpendicular to the tangent must pass through the tangent point.
Inference 2 A straight line passing through the tangent and perpendicular to the tangent must pass through the center of the circle.
The angle at which the vertex is on the circle, one side intersects the circle and the other side is tangent to the circle is called the tangent angle. It is the third angle related to the circle after the central angle and the peripheral angle. This angle must meet three conditions:
(1) The vertex is on the circle, that is, the vertex of the angle is the tangent point of a tangent of the circle;
(2) One side of the angle intersects the circle, that is, one side of the angle is the ray where a chord of the tangent point is located;
(3) The other side of the angle is tangent to the circle, that is, the other side of the angle is a ray with the tangent point as the endpoint. They are the criteria for judging whether an angle is a tangent angle, and all three are indispensable. For example, in the picture below, none of them are tangent angles;
(4) The tangent angle can be considered as a special case of the circumferential angle, that is, the angle formed when one side of the circumferential angle rotates around the vertex and is tangent to the circle. Therefore, the tangent angle has similar properties to the circumferential angle.
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