Circles MM2G3. Students will understand the properties of circles. b. Understand and use properties of central, inscribed, and related angles. d. Justify measurements and relationships in circles using geometric and algebraic properties.

Properties of a Circle A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. A segment whose endpoints are the center and any point on the circle is a radius. A chord is a segment whose endpoints are on a circle. A diameter is a chord that contains the center of the circle.

A secant is a line that intersects a circle in two points. A tangent is a line in the plane of a circle that intersects the circle in exactly one point, called the point of tangency. A tangent to a circle is perpendicular to the radius at the point of tangency.

secant tangent point of tangency

Tangent is perpendicular to a radius at the point of tangency. Tangent Theorems Tangent is perpendicular to a radius at the point of tangency. Create right triangles for problem solving.

Tangents of Curvature Do the 14-4 Enrichment of Tangents, Secant, and intercepted Arcs, # 1 – 4 Look at GPS files showing the fact that the point of tangency has to be on the line through the centers of the two circles. Assign the egg tangent project – due in a week

Warm-Up

Angles Inside of the Circle Have students make secant angles inside of a circle, measure the angles, discover the angle relationship. HW: page 214, # 5, 8, 9, 13, 15 HW: page 216, # 13

Warm Up

Angles Outside of the Circle Review: A secant line is a line that intersects a circle in two points. A tangent line intersects a circle at one point New: In a circle, two secants, one secant and one tangent, or two tangents can intersect outside of the circle to form an angle.

secant tangent point of tangency

Angles Outside of the Circle Use the circle template provided. Use your straight edge to construct an angle outside of the circle using two secants. Label the vertex of your angle and the points where the sides of the angle intersect the circle. How many arcs will two secants cut the circle into?

Angles Outside of the Circle Use your straight edge to construct an angle outside of the circle using one secant and one tangent. Label the vertex of your angle and the points where the sides of the angle intersect the circle. How many arcs will one secant and one tangent cut the circle into?

Angles Outside of the Circle Use your straight edge to construct an angle outside of the circle using two tangents. Label the vertex of your angle and the points where the sides of the angle intersect the circle. How many arcs will two tangents cut the circle into?

Angles Outside of the Circle Shade the arcs per the GPS file Write the equation needed to find the measure of the angle outside the circle for each example. What general rule do you notice? angle = ½(lg. arc – sm. arc)

Arcs and Angles There are three possible arc/angle situations. Vertex located ON THE CIRCLE. Vertex located IN THE CIRCLE. Vertex located OUTSIDE THE CIRCLE.

Vertex ON THE CIRCLE angle = ½ arc 250° 110° 125° 55°

Vertex IN THE CIRCLE = 120° angle = ½(arc + arc) 110° 130° = ½(240) = 120° 130° angle = ½(arc + arc)

Vertex OUTSIDE THE CIRCLE

Vertex OUTSIDE THE CIRCLE angle = ½(160 - 50) = ½(110) = 55° 160° 50° angle = ½(lg. arc – sm. arc)

Vertex ON THE CIRCLE angle = ½ arc Vertex IN THE CIRCLE angle = ½(arc + arc) Vertex OUTSIDE THE CIRCLE angle = ½(lg. arc – sm. arc)

Practice Page 214, all not assigned 4/18.