Lesson 10-R Chapter 10 Review

Objectives Review Chapter 10 material

Parts of Circles Circumference (Perimeter) –once around the outside of the circle; Formulas: C = 2πr = dπ Chord –segment with endpoints of the edge of the circle Radius –segment with one endpoint at the center and one at the edge Diameter –segment with endpoints on the edge and passes thru the center –longest chord in a circle –is twice the length of a radius Other parts –Center: is also the name of the circle –Secant: chord that extends beyond the edges of the circle –Tangent: a line (segment) that touches the circle at only one point

Arcs in Circles Arc is the edge of the circle between two points An arc’s measure = measure of its central angle All arcs (and central angles) have to sum to 360° If two arcs have the same measure then the chords that form those arcs have the same measure If a radius is perpendicular to a chord then it bisects the chord and the arc formed by the chord (example arc AED below) Major Arc (example: arc DAB) –measures more than 180° –more than ½ way around the circle Minor Arc (example: arc AED) –measures less than 180° –less than ½ way around the circle Semi-circle (example: arc EAB) –measures 180° –defined by a diameter A D C 120° BE is a diameter and AB = AD B E 120° 60°

Angles Associated with Circles Name Vertex Location SidesFormulaExample CentralCenterradii= measure of the arc  BCD = 110° InscribedEdgechords= ½ measure of the arc  BAD = 55° InteriorInsidechords= average of the vertical arcs  EVH = 73° ExteriorOutside Secants / Tangents = ½ (Big Arc – Little Arc) = ½ (Far Arc – Near Arc)  NVM = 30° V K L M N A D B C 110° minor arc BD = 110° E G F C 110° H 36° V minor arc FG = 110° minor arc EH = 36° minor arc LK = 10° minor arc NM = 70° C 70° 10°

Segments Inside/Outside of Circles Segments that intersect inside or outside the circle have the length of their parts defined by: J J K K L M M N J K T M LJ · JM = NJ · JK 3  8 = 6  4 JL · JN = JK · JM 5  12 = 4  15 JT · JT = JK · JM 6  6 = 3  12 Two Chords Inside a Circle Two Secants From Outside Point Secant & Tangent From Outside Point L N Inside the circle, it’s the parts of the chords multiplied together Outside the circle, it’s the outside part multiplied by the whole length O  W = O  W

Tangents and Circles Tangents and radii always form a right angle We can use the converse of the Pythagorean theorem to check if a segment is tangent The distance from a point outside the circle along its two tangents to the circle is always the same distance Example 1 Given: JT is tangent to circle C JC = 25 and JT = 20 Find the radius S T J C Example 2 Given: same radius as example 1 JC = 25 and JS = 16 Is JS tangent to circle C? JC² = JT² + TC² 25² = 20² + r² 625 = r² 225 = r² 15 = r JC² = JS² + SC² 25² = 16² + 15² 625 = ≠ 481 JS is not tangent

Equation of Circles A circle’s algebraic equation is defined by: (x – h)² + (y – k)² = r² where the point (h, k) is the location of the center of the circle and r is the radius of the circle Circles are all points that are equidistant (that is the distance of the radius) from a central point (the center)

Summary & Homework Summary: –A Homework: –study for the test