What is the Type of Tangents to Circles?

• Perpendicular to Radius: The radius drawn to the point of tangency is always perpendicular to the tangent. In other words, the angle between the radius and the tangent at the point of tangency is always \(90^\circ\).
• Equal Lengths: If two tangents are drawn from a single external point to a circle, they are congruent.

• The tangents are congruent.
• The segments joining the center of the circle and the points of tangency form two congruent angles.

Examples

Practice Questions:

1. If two tangents are drawn from an external point and they meet the circle at \( A \) and \( B \) respectively, and \( AB = 10 \text{ cm} \), what is the length of each tangent segment?
2. A tangent at \( T \) and a secant through points \( A \) and \( B \) intersect outside a circle at \( P \). If arc \( ATB \) measures \( 140^\circ \) and arc \( AB \) measures \( 60^\circ \), what is the measure of \( \angle PTB \)?
3. From an external point \( P \), two tangents are drawn to a circle of radius \( 6 \text{ cm} \). If the distance between \( P \) and the center of the circle is \( 10 \text{ cm} \), find the length of one of the tangent segments.

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1. \( 5 \text{ cm} \) for each segment (since the two tangents are congruent).
2. \( 40^\circ \) (half the difference of the measures of the intercepted arcs).
3. \( 8 \text{ cm} \) (using the Pythagorean theorem).

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