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Questions and Answers
What does Theorem 9-1 state?
What does Theorem 9-1 state?
- Tangents to a circle from a point are diameters.
- The line is tangent to the circle.
- The line is perpendicular to the radius at the endpoint.
- If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. (correct)
What is the Corollary for Theorem 9-1?
What is the Corollary for Theorem 9-1?
- Tangents to a circle from a point are congruent. (correct)
- The tangent to a circle is always longer than the radius.
- A diameter bisects the tangent at the point of contact.
- All tangents to a circle intersect at one point.
What does Theorem 9-2 indicate?
What does Theorem 9-2 indicate?
- A line is always perpendicular to the radius if it meets the circle.
- Chords can be tangent to the circle.
- All lines in the plane of a circle intersect at two points.
- The line is tangent to the circle if it is perpendicular to the radius. (correct)
What does the Arc Addition Postulate state?
What does the Arc Addition Postulate state?
Theorem 9-3 states that:
Theorem 9-3 states that:
What does Theorem 9-4 cover?
What does Theorem 9-4 cover?
According to Theorem 9-5, two minor arcs in the same or congruent circles are congruent if:
According to Theorem 9-5, two minor arcs in the same or congruent circles are congruent if:
Theorem 9-6 indicates that:
Theorem 9-6 indicates that:
What does Theorem 9-7 state about inscribed angles?
What does Theorem 9-7 state about inscribed angles?
Corollary 1 for Theorem 9-7 explains that:
Corollary 1 for Theorem 9-7 explains that:
Corollary 2 for Theorem 9-7 states that:
Corollary 2 for Theorem 9-7 states that:
Corollary 3 for Theorem 9-7 indicates that:
Corollary 3 for Theorem 9-7 indicates that:
According to Theorem 9-8, the measure of an angle formed by a chord and a tangent is:
According to Theorem 9-8, the measure of an angle formed by a chord and a tangent is:
Theorem 9-9 specifies that the measure of an angle formed by two chords that intercept inside a circle is:
Theorem 9-9 specifies that the measure of an angle formed by two chords that intercept inside a circle is:
What does Theorem 9-10 state?
What does Theorem 9-10 state?
Theorem 9-11 states that when two chords intersect inside a circle:
Theorem 9-11 states that when two chords intersect inside a circle:
According to Theorem 9-12:
According to Theorem 9-12:
Theorem 9-13 indicates that when a secant and a tangent segment are drawn to a circle from an external point:
Theorem 9-13 indicates that when a secant and a tangent segment are drawn to a circle from an external point:
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Study Notes
Tangents and Circles
- Theorem 9-1: A tangent line to a circle is perpendicular to the radius at the point where it touches the circle.
- Corollary for Theorem 9-1: Tangents drawn from a common external point are congruent.
- Theorem 9-2: A line that is perpendicular to the radius at its outer endpoint is tangent to the circle.
Arc Properties
- Arc Addition Postulate: The measure of an arc formed by two adjacent arcs equals the sum of their individual measures.
- Theorem 9-3: Two minor arcs are congruent in the same circle (or congruent circles) if and only if their central angles are congruent.
- Theorem 9-4: Congruent arcs imply congruent chords, and vice versa, within the same circle or congruent circles.
Chord and Diameter Relationships
- Theorem 9-5: A diameter perpendicular to a chord bisects both the chord and its corresponding arc.
- Theorem 9-6: Chords that are equidistant from the circle's center are congruent, while congruent chords maintain equal distance from the center.
Inscribed Angles
- Theorem 9-7: The measure of an inscribed angle is half the measure of its intercepted arc.
- Corollary 1 for Theorem 9-7: Inscribed angles that intercept the same arc are congruent.
- Corollary 2 for Theorem 9-7: An angle inscribed in a semicircle is always a right angle.
- Corollary 3 for Theorem 9-7: The opposite angles of a quadrilateral inscribed in a circle are supplementary.
Angles Formed by Chords and Tangents
- Theorem 9-8: The measure of an angle formed by a chord and a tangent is half the measure of the intercepted arc.
- Theorem 9-9: The angle formed by two intersecting chords inside a circle equals half the sum of the measures of the intercepted arcs.
- Theorem 9-10: The angle formed by two secants, two tangents, or a secant and tangent from an external point is half the difference of the intercepted arcs' measures.
Segment Products
- Theorem 9-11: When two chords intersect within a circle, the product of the segments of one chord equals the product of the segments of the other chord.
- Theorem 9-12: The product of the entire secant segment and its external segment equals the product of the other secant segment and its external segment for segments drawn from an external point.
- Theorem 9-13: The product of a secant segment with its external segment equals the square of the tangent segment drawn from the same external point.
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