Chapter 10: Circles Geometry H

10.1 Circles and Circumference Name a circle by the letter at the center of the circle Diameter- segment that extends from one point on the circle to another point on the circle through the center point Radius- segment that extends from one point on the circle to the center point Chord- segment that extends from one point on the circle to another point on the circle Diameter=2 x radius (d=2r) Circumference: the distance around the circle C=2πr or C= πd

Circle X Diameter- Radius- Chord- A chord B diameter E C X radius D

Name the circle 2. Name the radii 3. Identify a chord 4. Identify a diameter

Circles can intersect at 2, 1 or no points.

Find the circumference of a circle to the nearest hundredth if its radius is 5.3 meters. b. Find the diameter and the radius of a circle to the nearest hundredth if the circumference of the circle is 65.4 feet.

10.2 Measuring Angles and Arcs Central angle: an angle with a vertex at the center of the circle. Semi-circle: half the circle (180 degrees) Minor arc: less than 180 degrees Name with two letters Major arc: more than 180 degrees Name with three letters Minor arc = central angle Arc length:

Sum of Central Angles = 360 B Minor arc Minor arc Minor arc = AB or BC Semicircle = ABC or CDA Major arc = ABD or CBD AB + BC = 180 C Central angle X A D Semicircle

Find the value of x.

Find x and angle AZE

Find the measure of each minor arc.

10.3 Arcs and Chords If two chords are congruent, then their arcs are also congruent If a radius or diameter is perpendicular to a chord, it bisects the chord and its arc If two chords are equidistant from the center of the circle, the chords are congruent In inscribed quadrilaterals, the opposite angles are supplementary

A If FE=BC, then arc FE = arc BC Quad. BCEF is an inscribed polygon – opposite angles are supplementary angles B + E = 180 & angles F + C = 180 Diameter AD is perpendicular to chord EC – so chord EC and arc EC are bisected B F C E D

*You can use the pythagorean theorem to find the radius B You will need to draw in the radius yourself E *You can use the pythagorean theorem to find the radius when a chord is perpendicular to a segment from the center XE = XF so chord AB = chord CD because they are equidistant from the center A X C F D

In the circle below, diameter QS is 14 inches long and chord RT is 10 inches long. Find VU.

Some more challenging Q’s Find the distance from the center of a circle to a chord that is 80m long if the diameter of the circle is 82m long. Two circles intersect and have a common chord that is 16cm long. The centers are 21cm apart. If the radius is 10cm, find the radius of the other circle.

10.4 Inscribed Angles Inscribed angle: an angle inside the circle with sides that are chords and a vertex on the edge of the circle Inscribed angle = ½ intercepted arc Intercepted arc = 2* inscribed angle An inscribed right angle, always intercepts a semicircle If two or more inscribed angles intercept the same arc, they are congruent In inscribed quadrilaterals, the opposite angles are supplementary

Intercepted arc: has endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle.

<BAC, <CAD, <DAE, <BAD, <BAE, <CAE Inscribed angles: <BAC, <CAD, <DAE, <BAD, <BAE, <CAE A Ex: Angle DAE = ½ arc DE X B C E D

Inscribed angle BAC intercepts a semicircle- so angle BAC =90 Inscribed angles GDF and GEF both intercept arc GF, so the angles are congruent B C D F E G

A. Find mX.

Refer to the figure. Find the measure of angles 1, 2, 3 and 4.

ALGEBRA Find mR.

ALGEBRA Find mI.

ALGEBRA Find mB.

ALGEBRA Find mD.

The insignia shown is a quadrilateral inscribed in a circle The insignia shown is a quadrilateral inscribed in a circle. Find mS and mT.

More problems A 10 by 24 rectangle is inscribed in a circle. Find the radius of the circle. Find the exact area of a circle whose diameter is 15 inches.

10.5 Tangents Tangent: a line that shares only one point with a circle and is perpendicular to the radius or diameter at that point. Point of tangency: the point that a tangent shares with a circle Two lines that are tangent to the same circle and meet at a point, are congruent from that point to the points of tangency Common Tangent: line, ray or segment that is tangent to two circles in the same plane.

Lines AC and AF are tangent to circle X at points B and E respectively -B and E are points of tangency Radius XB is perpendicular to tangent AC at the point of tangency AE and AB are congruent because they are tangent to the same circle from the same point X C F E B A

More Walk Around Problems

Common Tangents

How to deal with common tangent problems Draw an appropriate diagram Draw a segment connecting their centers Draw the radii to the points of contact Through the center of the smaller circle, draw a line parallel to the common tangent Extend any radius if necessary to obtain right triangles and rectangle Use the Pythagorean theorem and properties of a rectangle

Example

10.6 Secants, Tangents, and Angle Measures Secant and Tangent Interior angle = ½ intercepted arc Two Secants: Interior angle = ½ (sum of intercepted arcs) Two Secants Exterior angle = ½ (far arc – close arc) Two Tangents

2 Secants/chords: Angle 1 = ½ (arc AD + arc CB) Angle 2 = ½ (arc AC + arc DB) C B 2 1 A D

Secant ED intersects tangent FC at a point of tangency (point F) Angle 1 = ½ arc FE Angle 2 = ½ (arc EA – arc FB) E 1 F B A 2 D C

A. Find x.

B. Find x.

C. Find x.

A. Find mQPS.

B.

A.

B.

10.7 Special Segments in a Circle Two Chords seg1 x seg2 = seg1 x seg2 Two Secants outer segment x whole secant = outer segment x whole secant Secant and Tangent outer segment x whole secant = tangent squared *Add the segments to get the whole secant

10.7 Special Segments in a Circle Chord segments: when two chords intersect inside a circle and divide each other into two segments. Secant segment: a segment of a secant line that has exactly one endpoint on the circle. External secant segment: a secant segment that lies in the exterior of the circle. Tangent segment: segment of a tangent with one endpoint on the circle

E D A F 2 chords: AO x OB = DO x OC 2 secants: EF x EG = EH x EI H O C I B G

A D Secant and Tangent: AD x AB = (AC)2 C B

A. Find x.

B. Find x.

A. Find x.

B. Find x.

Find x.

Find x.

LM is tangent to the circle. Find x. Round to the nearest tenth.

Find x. Assume that segments that appear to be tangent are tangent.

10.8 Equations of Circles