6.1 Circles and Related Segments and Angles • Circle - set of all points in a plane that are a fixed distance from.
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6.1 Circles and Related Segments and Angles • Circle - set of all points in a plane that are a fixed distance from a given point (center). • Radius – segment from the center to the edge of the circle. All radii of a circle are the same length. • Concentric circles - circles that have the same center. r 6.1 Circles and Related Segments and Angles • Definition: chord – segment that joins 2 points of the circle. • Theorem: A radius that is perpendicular to a chord bisects the chord. 6.1 Circles and Related Segments and Angles AB is a part of a circle • Definition: Arc – that connects points A and B – Minor arc (<180) – Semicircle (180) – Major arc (> 180) • Sum of all the arcs in a circle = 360 6.1 Circles and Related Segments and Angles • Central angle - vertex is the center of the circle and sides are the radii of the circle. 6.1 Circles and Related Segments and Angles • Postulate 16: m(BAC) m( BC) B A C Congruent arcs – arcs with equal measure. • Postulate 17: (Arc-addition postulate) If B lies between A and C, then m( AB) m( BC) m( ABC) 6.1 Circles and Related Segments and Angles • Inscribed angle – vertex is a point on the circle and the sides are chords of the circle. A • The measure of an inscribed angle is ½ the measure of its intercepted arc. m(ACB) 12 m( AB) C B 6.1 Circles and Related Segments and Angles m( AB) 80 and m( AC) 130 • Example: Given circle with center O, What is mAOB, mACB, and m(CB ) ? C O A B 6.2 More Angle Measures in a Circle • Tangent - a line that intersects a circle at exactly one point. • Secant - a line, segment, or ray that intersects a circle at exactly 2 points. 6.2 Inscribed/Circumscribed Polygons • Polygon inscribed in a circle - its vertices are points on the circle. Note: The circle is circumscribed about the polygon. • Polygon circumscribed about a circle - all sides of the polygon are segments tangent to the circle. Note: The circle is inscribed in the polygon. 6.2 More Angle Measures in a Circle • Given an angle formed by intersecting chords: 1 m1 2 m AD m BC note: 1 is not a central angle because E is not the center A 1 D B E C 6.2 More Angle Measures in a Circle C B A D • Given an angle formed by intersecting secants: 1 mA 2 m CE m BD E 6.2 More Angle Measures in a Circle B D A C • Given an angle formed by intersecting tangents: 1 mA 2 m BDC m BC 6.2 More Angle Measures in a Circle D • Given an angle formed by an intersecting secant B and tangent: 1 mA m BD m BC 2 C A 6.2 More Angle Measures in a Circle • The radius drawn to a tangent at the point of tangency is perpendicular to the tangent. • Given a line tangent to a circle and a chord intersecting at the point of tangency: B 1 mBAC 2 m AB A C 6.3 Line and Segment Relationships in the Circle • Given an angle formed by intersecting chords: AE EC BE ED A 1 D B E C 6.3 Line and Segment Relationships in the Circle C B A D • Given an angle formed by intersecting secants: AB AC AD AE E 6.3 Line and Segment Relationships in the Circle B D A C • Given an angle formed by intersecting tangents: AB AC 6.3 Line and Segment Relationships in the Circle D C B • Given an angle formed by an intersecting secant and tangent: AB AC AD 2 A 6.4 Inequalities for the circle • m1 m2 m( AB) m(CD) A D B 1 2 C 6.4 Inequalities for the circle • Given a circle with center O and distances to the center are OP and OQ AB CD OP OQ A P O D B C Q 6.4 Inequalities for the circle • Given two chords in a circle: AB CD m( AB) m(CD) A B C D 6.4 Summary: Inequalities for the circle Chords Central Angles AB < CD mAOB < mCOD AB > CD mAOB > mCOD A Distances to Center PO>QO PO<QO P B C O Q D Arcs m( AB) m(CD) m( AB) m(CD) 6.5 Locus of Points • A locus of points - set of all points that satisfy a given condition. – Example: Locus of points that are a fixed distance (r) from a point (p) – a circle with center p and radius r – Example: Locus of points that are equidistant from 2 fixed points – perpendicular bisector of the segment between them. 6.5 Locus of Points • More Examples: – Locus of points in a plane that are equidistant from the sides of an angle – the ray that bisects the angle. – Locus of points in space that are equidistant from two parallel planes – a plane parallel to both planes but midway in between. – Locus of all vertices of right triangles having a hypotenuse AB – circle of diameter AB. 6.6 Concurrence of Lines • Concurrent - A number of lines are concurrent if they have exactly one point in common. • The 3 angle bisectors of the angles of a triangle are concurrent. • The 3 perpendicular bisectors of the sides of a triangle are concurrent. • The 3 altitudes of the sides of a triangle are concurrent. (point of concurrence is called the orthocenter) 6.6 Concurrence of Lines • The 3 medians of a triangle are concurrent at a point that is 2/3 the distance from any vertex to the midpoint of the opposite side. • Definition: – Centroid: point of concurrence of the 3 medians of a triangle 7.1 Areas and Initial Postulates • Postulate 21 – Area of a rectangle = length width w l 7.1 Areas and Initial Postulates • Area of a triangle = ½ base height h b 7.1 Areas and Initial Postulates • Area of a parallelogram = base height h b h b 7.1 Areas and Initial Postulates Polygon Area Square s2 Rectangle lw Parallelogram bh Triangle ½ (b h) 7.2 Perimeters and Areas of Polygons Polygon Triangle Perimeter a + b + c (3 sides) Quadrilateral a + b + c + d (4 sides) Polygon Sum of all lengths of sides Regular Polygon #sides length of a side 7.2 Perimeters and Areas of Polygons • Heron’s formula – If 3 sides of a triangle have lengths a, b, and c, then the area A of a triangle is given by: A s ( s a)(s b)(s c) where s is thesemi - perimeters 12 (a b c) • Why use Heron’s formula instead of A = ½ bh? 7.2 Perimeters and Areas of Polygons • Area of a trapezoid – A = ½ h (b1 + b2) b2 h b1 7.2 Perimeters and Areas of Polygons • Area of a kite (or rhombus) with diagonals of lengths d1 and d2 is given by A = ½ d1 d2 d1 d2 7.2 Perimeters and Areas of Polygons • The ratio of the areas of two similar triangles equals the square of the ratio of the lengths of any two corresponding sides. A1 a1 A2 a2 a1 A1 a2 A2 2 7.3 Regular Polygons and Area • Center of a regular polygon – common center for the inscribed and circumscribed circles. • Radius of the regular polygon – joins the center of the regular polygon to any one of the vertices. Center Radius 7.3 Regular Polygons and Area • Apothem – is a line segment from the center to one of the sides and perpendicular to that side. • Central angle – is an angle formed by 2 consecutive radii of the regular polygon. • Measure of the central angle: C 360 n central angle apothem 7.3 Regular Polygons and Area • Area of a regular polygon with apothem length a and perimeter P is: A 12 aP where P = number of sides length of side P ns s apothem 7.4 Circumference and Area of a Circle r • Circumference of a circle: m C = d = 2r l 22/7 or 3.14 • Length of an arc – the length l of an arc whose degree measure is m is given by: m l C 360 7.4 Circumference and Area of a Circle • Limits: Largest possible chorddiameter • For regular polygons ( A 12 aP) as n – apothem (a) r (radius) – perimeter (P) C(circumference) = 2r • Area of a circle – A r (2 r ) r 1 2 2 7.5 More Area Relationships in the Circle • Sector – region in a circle bounded by 2 radii and their intercepted arc Asec tor m m Acircle r2 360 360 r m sector 7.5 More Area Relationships in the Circle • Segment of a circle – region bounded by a chord and its minor/major arc Note: For problems use: Atriangle Asegment Asec tor 7.5 More Area Relationships in the Circle • Theorem: Area of a triangle = ½ rP where P = perimeter of the triangle and r = radius of the inscribed triangle Note: This can be used to find the radius of the inscribed circle.