Geometry Notes: Tangents, Arcs, Chords 9.1
Circle – set of points in a plane at a given distance from a given point in that plane.
Center – the given point of the circle
Radius – the line segment from the center of the circle to a point on the circle. (half the diameter)
Chord – a segment whose endpoints lie on a circle
Secant – a line that contains a chord.
Diameter – a chord that contains the center of the circle. The largest chord in a circle. Diameter is twice the radius.
Tangent – a line in the plane of a circle that intersects the circle in exactly one point.
Point of tangency -- The point where the tangent and the circle intersect
Sphere – set of all points in space at a distance r (radius) from the center point.
Congruent circles (or spheres) are circles (or spheres) that have congruent radii.
Concentric Circles – circles that lie in the same plane and have the same center.
Concentric spheres – spheres that have the same center.
A polygon is inscribed in a circle and the circle is circumscribed about the polygon when each vertex of the polygon lies on the circle.
9.2 Geometry Tangents
If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.
Tangents to a circle from a point are congruent.
If a line in the plane of a circle is perpendicular to a radius at the outer endpoint, then the line is tangent to the circle.
When each side of a polygon is tangent to a circle, the polygon is said to be circumscribed about the circle and the circle is inscribed in the polygon.
A line that is tangent to each of two coplanar circles is called a common tangent
A common internal tangent intersects the segment joining the centers.
A common external tangent does not intersect the segment joining the centers.
Tangent circles are coplanar circles that are tangent to the same line at the same point.
9.3 Geometry Arcs and Central Angles
Central Angle – an angle in a circle with its vertex at the center of the circle.
Minor arc -- part of a circle named by its 2 endpoints. A minor arc has a measure of less than 180 degrees.
Major arc – part of a circle that has a measure more than 180 degrees. It is named by three letters, 2 of the letters are its endpoints.
Semicircle – an arc whose measure is exactly 180 degrees.
Adjacent arcs -- arcs that have exactly one point in common.
The measure of the arc formed by two adjacent acrs is the sum of the measures of these two arcs.
Congruent arcs – arcs in the same circle or in congruent circles that have equal measures.
In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent.
9.4 Geometry Arcs and Chords
In the same circle or in congruent circles
(1) Congruent arcs have congruent chords
(2) Congruent chords have congruent arcs.
A diameter that is perpendicular to a chord bisects the chord and its arc.
In the same circle or in congruent circles:
(1) Chords equally distant from the center (or centers) are congruent
(2) Congruent chords are equally distant from the center (or centers)
9.5 Geometry Inscribed Angles
Inscribed angle – an angle whose vertex is on a circle and whose sides are contain chords of the circle.
The measure of an inscribed angle is equal to half the measure of its intercepted arc.
If two inscribed angles intercept the same arc, then the angles are congruent.
An angle inscribed in a semicircle is a right angle.
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.
9.6 Geometry Other Angles
The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.
The measure of an angle formed by two secants, two tangents or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measure of the intercepted arcs.
9.7 Geometry Circles and Lengths of Segments
When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord.
When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external secant segment equals the product of the other secant segment and its external secant segment.
When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external secant segment is equal to the square of the tangent segment.