The Bronx High School of Science, Mathematics Department. Trigonometry (M$6) Mr. J. Fox, Instructor M$6 CIRCLE REVIEW SHEET

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1 The ronx High School of Science, Mathematics epartment Valerie Reidy, rincipal Rosemarie Jahoda,.. Mathematics Trigonometry (M$6) Mr. J. Fox, Instructor M$6 IRLE REVIEW SHEET Note: This review sheet was originally created by Mr. Tamhane. He deserves all credit for the work (and interesting methods to remember some circle theorems!) Here are the essentials you might need to know about circles for the regents exam! First, a few definitions so you know what we re talking about: entral ngle n angle whose vertex coincides with the center of the circle. Inscribed ngle n angle whose vertex coincides with a point on the circle, and whose rays intersect the circle at two points distinct from the vertex. O is a central angle is an inscribed angle O is NOT an inscribed angle rc continuous portion of a circle. Notation: rc Measure The number of degrees of the central angle that intercepts an arc. Notation: m NOTE THT R MESURE IS IFFERENT FROM R LENGTH ( measurement of distance).

2 Useful Formulas: The measure of an arc = the measure of the central angle that intercepts it Or in mathematical terms (using the diagram above) m = m O There are certain lines/line segments that you may come across when dealing with circles. Here s a quick review: hord line segment whose endpoints are at two distinct points on the circle. Secant line that intersects a circle at two distinct points. This is similar to a chord. Without getting bogged down in mathematical definitions, the biggest difference that is relevant to you is that a secant line extends beyond the confines of the inside of the circle. Tangent line that intersects the circle at exactly one point. KEY ROERTY: tangent line forms a right angle with a radius of the circle. is a chord. EF is a secant line. O is a tangent line F E

3 The following is an overview of the types of questions that may appear on the Math Regents exam: ngle between two lines/segments whose vertex is ON the circle Inscribed ngle If an angle s vertex is on a circle, then some arc is going to be intercepted by the angle (aka, it will be between the rays of the angle). Take half the measure of that arc to find the measure of the angle. ngle between a tangent line and a chord Or in mathematical terms (referring to either diagram): 1 m = (m ) ngle between two lines/segments whose vertex is INSIE the circle ngle between two chords When two chords intersect in a circle, they break the circle into 4 arcs. To find the angle between them, take the arc intercepted by the angle and the arc on the opposite side of the circle. Find the sum between their measures, and divide by two. Or in mathematical terms (according to the diagram): m = m + m

4 ngle between two lines/segments whose vertex is OUTSIE the circle ngle between two secants If an angle s vertex is outside of a circle (and its rays intersect the circle in two distinct places), then the angle intercepts two arcs. If you take the positive difference of these arc measures and divide by two, you will find the measure of the angle. ngle between a secant and tangent Or in mathematical terms (referring to any of the diagrams to the left or below): m m m = ngle between two tangents Finding Segment Lengths There are certain equations which summarize the relationship between the lengths of segments. Lengths of hord Segments When two chords intersect, each chord is broken into two parts. If you multiply the lengths each segment of a chord, that product will equal the product of the other chords segments. ()() = ()() (Segment of 1 st chord)*(other segment of 1 st chord) = Segment of nd chord)*(other segment of nd chord)

5 Lengths of Secants When analyzing the lengths of two secants, notice how both line segments have parts that are outside of the circle. If you take the length of the whole secant and multiply it by the legnth of its external part, this will be equal to the whole other secant times the external part of the other secant. ()() = ()() (Whole 1 st secant)*(external part of 1 st secant) = (Whole nd secant)*(external part of nd secant) This could be considered the WE-WE theorem. Lengths of Tangents and Secants The length of the whole secant times its external part is equal to the length of the tangent segment squared. ()() = () (Whole Secant)*(External part of Secant) = (Tangent) I call this the WET theorem.

Unit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with

Unit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with endpoints on the circle. Diameter - A chord which passes through