Circles Chapter 10.

Circles Chapter 10

Lines and Segments that Intersect Circles
I can identify special segments and lines, and use properties of tangents.

Vocabulary (page 280 in Student Journal) circle: the set of points in a plane that are equidistant from a given point center: the point in a circle equidistant from all the points on the circle radius: a segment whose endpoints are the center and any point on the circle chord: a segment whose endpoints are on the circle diameter: a chord that contains the center

secant: a line that intersects a circle in 2 points tangent: a line in the plane of a circle that intersects the circle in exactly 1 point point of tangency: the point where a tangent intersects the circle tangent circles: coplanar circles that intersect a 1 point concentric circles: coplanar circles that have a common center common tangent: a line or segment that is tangent to 2 coplanar circles

Core Concepts Tangent Line to Circle Theorem In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle. External Tangent Congruence Theorem Tangent segments from a common external point are congruent.

Examples (space on pages 280 and 281 in Student Journal) a) Identify a diameter, radius, chord, secant and tangent in the diagram.

Solution diameter: segment RP radius: segment RO, segment PO, or segment QO chord: segment PQ secant: line PQ tangent: line MN

Determine how many common tangents the circles have in the diagram. b) c) d)

Solutions b) 0 c) 1 d) 4

e) Is segment ST tangent to circle P?

Solution e) no

f) Find the radius if point P is the point of tangency.

Solution f) 15 units

g) Find the value of x, if segment JH is tangent to circle L at H and if segment JK is tangent to circle L at K.

Solution g) x = 12

Finding Arc Measures I can find arc measures and identify congruent arcs.

Finding Arc Measures Vocabulary (page 285 in Student Journal) central angle: an angle whose vertex is the center of the circle minor arc: for a central angle less than 180 degrees, the points on the circle inside the rays forming the central angle major arc: for a central angle greater than 180 degrees, the points on the circle inside the rays forming the central angle semicircle: an arc with endpoints that are the endpoints of a diameter

Finding Arc Measures measure of minor arc: the measure of the central angle that forms the arc measure of major arc: the difference of 360 degrees and the measure of the related minor arc adjacent arcs: 2 arcs of the same circle that intersect at exactly 1 point congruent circles: circles that can be mapped onto 1 another using a rigid motion congruent arcs: arcs that have the same measure and are arcs of the same circle or congruent circles similar arcs: arcs that have the same measure

Finding Arc Measures Core Concepts (page 286 in Student Journal) Arc Addition Postulate The measure of an arc formed by 2 adjacent arcs is the sum of the measures of the 2 arcs. Congruent Circles Theorem Two circles are congruent circles if and only if they have the same radius.

Finding Arc Measures Congruent Central Angles Theorem In the same circle, or in congruent circles, 2 minor arcs are congruent if and only if their corresponding central angles are congruent. Similar Circles Theorem All circles are similar.

Finding Arc Measures Examples (space on pages 285 and 286 in Student Journal) Find the measure of each arc of circle C, where segment AB is the diameter. a) arc AD b) arc DAB c) arc BDA

Finding Arc Measures Solutions a) 65 degrees b) 245 degrees c) 180 degrees

Finding Arc Measures Find the measure of each arc. d) arc SQ e) arc RPQ f) arc PRS

Finding Arc Measures Solutions d) 125 degrees e) 270 degrees f) 215 degrees

Finding Arc Measures Determine if the red arcs in the diagram are congruent. Explain. g) h) i)

Finding Arc Measures Solutions g) No, circles are not congruent h) Yes, same arc measure and same circle i) Yes, congruent circles and same arc measure

Using Chords I can use chords of circles to find lengths and arc measures.

Using Chords Core Concepts (pages 290 and 291 in Student Journal) Congruent Corresponding Chords Theorem In the same circle, or in congruent circles, 2 minor arcs are congruent if and only if their corresponding chords are congruent. Perpendicular Chord Bisector Theorem If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

Using Chords Perpendicular Chord Bisector Converse If 1 chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter. Equidistant Chords Theorem In the same circle, or in congruent circles, 2 chords are congruent if and only if they are equidistant from the center.

Using Chords Examples (space on pages 290 and 291 in Student Journal) a) The circles in the diagram are congruent. Find the measure of arc FG.

Using Chords Solution a) 120 degrees

Using Chords Use the diagram to answer the following. b) KH c) measure of arc HLK

Using Chords Solutions b) 28 c) 196 degrees

Using Chords d) Use the information in the diagram to find the radius of circle E.

Using Chords Solution d) 20 units

Inscribed Angles and Polygons
I can use inscribed angles and inscribed polygons.

Vocabulary (page 295 in Student Journal) inscribed angle: an angle whose vertex in on a circle and whose sides contain chords of the circle intercepted arc: an arc that lies between 2 lines, rays or segments subtend: when the endpoints of a chord or arc lie on the sides of an inscribed angle inscribed polygon: all the vertices of the polygon lie on the circles circumscribed circle: the circle that contains all the vertices of a polygon

Core Concepts (pages 295 and 296 in Student Journal) Measure of an Inscribed Angle Theorem The measure of an inscribed angle is ½ the measure of its intercepted arc. Inscribed Angles of a Circle Theorem If 2 inscribed angles of a circle intercept the same arc, then the angles are congruent.

Inscribed Right Triangle Theorem If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if 1 side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. Inscribed Quadrilateral Theorem A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

Examples (space on pages 295 and 296 in Student Journal) Find the indicated measures using the diagram. a) measure of arc DG b) measure of angle G

Solutions a) 80 degrees b) 29 degrees

Find the indicated measures in the diagram. c) measure of arc HML d) measure of angle HJL

Solutions c) 218 degrees d) 109 degrees

e) Find the measure of angle B in the diagram.

Solution e) 68 degrees

Find the value of each variable in the diagrams. f) g)

Solutions f) x = 18 g) a = 23, b = 115

Angle Relationships in Circles
I can find angle and arc measures and use circumscribed angles.

Vocabulary (page 300 in Student Journal) circumscribed angle: an angle whose sides are tangent to a circle

Core Concepts (pages 300 and 301 in Student Journal) Tangent and Intersected Chord Theorem If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is ½ the measure of its intercepted arc. Angles Inside the Circle Theorem If 2 chords intersect inside a circle, then the measure of each angle is ½ the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Angles Outside the Circle Theorem If a tangent and a secant, 2 tangents, or 2 secants intersect outside a circle, then the measures of the angle formed is ½ the difference of the measures of the intercepted arcs. Circumscribed Angle Theorem The measure of a circumscribed angle is equal to 180 degrees minus the measure of the central angle that intercepts the same arc.

Examples (space on pages 300 and 301 in Student Journal) Find the measure of the red angle or arc, if line m is tangent to the circle. a) b)

Solutions a) 120 degrees b) 310 degrees

Find the value of x in the diagram. c) d)

Solutions c) x = 92 d) x = 30

Find the value of x. e) f)

Solutions e) x = 56 f) x = 69

Segment Relationships in Circles
I can use segments of chords, tangents, and secants.

Vocabulary (page 305 in Student Journal) segments of a chord: when a chord is divided into 2 segments tangent segment: a segment that is tangent to a circle at an endpoint secant segment: a segment that contains a chord of a circle and has exactly 1 endpoint outside the circle external segment: the part of a secant segment that is outside the circle

Core Concepts (pages 305 and 306 in Student Journal) Segments of Chords Theorem If 2 chords intersect in the interior of a circle, then the product of the lengths of the segments of 1 chord is equal to the product of the lengths of the segments of the other chord. Segments of Secants Theorem If 2 secant segments share the same endpoint outside of a circle, then the product of the lengths of 1 secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

Segments of Secants and Tangents Theorem If a secant and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

Examples (space on pages 305 and 306 in Student Journal) Use the diagram to find the indicated measures. a) AB b) PQ

Solutions a) 14 b) 13

c) Find the value of x.

Solution c) x = 5

d) Find WX.

Solution d) 18

e) Find the radius of the circle.

Solution e) 4 units

Circles in the Coordinate Plane
I can write and graph equations of circles.

Vocabulary (page 310 in Student Journal) standard equation of a circle: (x – h)2 + (y – k)2 = r2, where (h, k) are the coordinates of the center of the circle and r is the radius

Examples (space on page 310 in Student Journal) Write the standard equation of the circle. a) b) center at the origin with a radius of 3.5 c) Center at (1, 4) with the point (4, 1) on the circle

Solutions a) (x + 5)2 + (x – 4)2 = 16 b) x2 + y2 = c) (x – 1)2 + (y – 4)2 = 18

d) The equation of a circle is x2 + y2 – 2x + 6y – 6 = 0. Find the center and the radius.

Solution d) center: (1, -3), radius: 4

e) Does the point (3, √7) lie on the circle centered at the origin and containing the point (1, 4)? Explain.

Solution e) no, not the same distance away from the origin as (1, 4)

Circles Chapter 10.

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