Tangents and Secants to a Circle

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A tangent to a circle is a line that touches the circle at exactly one point. This point is called the point of contact. The tangent does not intersect the circle at any other point.

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A secant to a circle is a line that intersects the circle at exactly two distinct points. The line segment between these two points on the circle is called a chord.

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Theorem 9.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact. In other words, if XY is tangent to circle with center O at point P, then OP ⊥ XY.

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No tangents can be drawn to a circle from a point inside the circle. All lines through an interior point will intersect the circle at two points, making them secants.

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Exactly one tangent can be drawn to a circle from a point on the circle. This tangent passes through the point and is perpendicular to the radius at that point.

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Exactly two tangents can be drawn to a circle from an external point. These tangents will have equal lengths from the external point to their respective points of contact.

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Theorem 9.2: The lengths of tangents drawn from an external point to a circle are equal. If PA and PB are tangents from external point P to a circle, then PA = PB.

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Using Pythagoras theorem in the right triangle formed: Tangent² = Distance² - Radius² Tangent² = 10² - 6² = 100 - 36 = 64 Tangent = √64 = 8 cm…