CIRCLES

Circle P The set of all points in a plane that are equidistant from a given point, called the CENTER of the circle. The word circle is derived from the Latin word circus, which means “ring” or “racecourse.”

Naming Circles P

Interior: Exterior: Interior/Exterior Points inside the circle Points outside the circle G I F On the circle

Radius/Chord/Diameter B A segment that has the center as one endpoint and a point on the circle as the other endpoint. A chord that passes through the center of the circle. A segment whose endpoints are ON the circle. P E C Chord: Diameter:

Circumference The circumference of a circle is the distance around around the circle. C = 2πr or C = πd 5 P So the circumference is … 10π

A circular swimming pool with a 16-ft diameter will be enclosed in a circular fence 4 ft from the pool. What length of fencing material is needed? Round your answer to the next whole number. Draw a diagram of the situation. The pool and the fence are concentric circles. The diameter of the pool is 16 ft, so the diameter of the fence is 16 + 4 + 4 = 24 ft. Use the formula for the circumference of a circle to find the length of fencing material needed. C = d Formula for the circumference of a circle C = (24) Substitute. C 75.36 Simplify. About 76 ft of fencing material is needed.

Tangent: Secant: A LINE that intersects a circle at exactly one point. Secant/Tangent Tangent: Secant: A LINE that intersects a circle at two points. A LINE that intersects a circle at exactly one point. W X A B

How are chords, radii, and diameters different from secants and tangents? Chords, radii, and diameters are segments. Secants and tangents are lines. A secant can contain a chord. A chord can be a diameter. A diameter is two radii forming opposite rays. P C B

Secant/Tangent Tangent: The point at which the line intersects the circle is called the point of tangency. Point Y below B Y

Common Tangent: A line that is tangent to two circles Common Tangents B Common Tangent: A line that is tangent to two circles

Common Internal Tangent: Common External Tangent: Common Tangents R S U T A B B A C D Common Internal Tangent: Common External Tangent: A common tangent that intersects the segment that joins the centers of the circles A common tangent that does not intersect the segment that joins the centers of the circles.

So, how can two CIRCLES intersect? Circle Intersections So, how can two CIRCLES intersect? once not at all twice all points Congruent circles have congruent radii (or congruent diameters). Concentric circles have the same center.

Central angle: Sum of Central Angles: Central Angles/Arcs Central angle: An angle whose vertex is at the center of the circle. In the diagram, this would be . A B P Sum of Central Angles: The sum of the measures of the central angles of a circle with no interior points in common is 360. C

A researcher surveyed 2000 members of a club to find their ages A researcher surveyed 2000 members of a club to find their ages. The graph shows the survey results. Find the measure of each central angle in the circle graph. Because there are 360° in a circle, multiply each percent by 360 to find the measure of each central angle. 65+ : 25% of 360 = 0.25 • 360 = 90° 45–64: 40% of 360 = 0.4 • 360 = 144° 25–44: 27% of 360 = 0.27 • 360 = 97.2° Under 25: 8% of 360 = 0.08 • 360 = 28.8°

Arc: Minor Arc: Major Arc: Central Angles/Arcs Arc: A part of a circle that consists of two points called endpoints and all the points in between A Major Arc:   Minor Arc:   D B P C

But, what if the arc has a measure of exactly 180 degrees? Semicircles But, what if the arc has a measure of exactly 180 degrees? What is it called then?….. X Y P Z Then, the arc is called a semicircle. For a semicircle, the endpoints of the arc will be the endpoints of a diameter. is a semicircle.  

Naming Arcs Minor arcs are usually named using 2 points Major arcs and semicircles are named using 3 points Arc names have the arc symbol over the point names A   B P   C

Identify the minor arcs, major arcs, and semicircles in P with point A as an endpoint. Two semicircles in the diagram have point A as an endpoint, ADB and AEB. Minor arcs are smaller than semicircles. Two minor arcs in the diagram have point A as an endpoint, AD and AE. Major arcs are larger than semicircles. Two major arcs in the diagram have point A as an endpoint, ADE and AED.

Measuring Arcs Minor arc measures are equal to their corresponding central angle.     A Major arc measures are equal to 360 minus their corresponding central angle.   B P     C

Arcs of the same circle that have exactly one point in common. Adjacent Arcs Arcs of the same circle that have exactly one point in common.   B A C P

Arc Addition Postulate The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.   B A C P

Find mXY and mDXM in C. . mXY = mXD + mDY Arc Addition Postulate mXY = m XCD + mDY The measure of a minor arc is the measure of its corresponding central angle. mXY = 56 + 40 Substitute. mXY = 96 Simplify. mDXM = mDX + mXWM Arc Addition Postulate mDXM = 56 + 180 Substitute. mDXM = 236 Simplify.

Congruent Arcs Congruent Arcs In the same circle, or in congruent circles, two arcs are congruent if and only if their central angles are congruent. Which of the diagrams show congruent arcs? 80 45 45 80

The arc that lies in the interior of an inscribed angle An angle whose vertex is ON the circle and whose sides are chords of the circle. A B Intercepted Arc: The arc that lies in the interior of an inscribed angle   C

Measure Inscribed Angle = ½ Measure of Intercepted Arc     35° C B

Find the values of x and y. x = mDEF Inscribed Angle Theorem 1 2 x = (mDE + mEF) Arc Addition Postulate 1 2 x = (80 + 70) Substitute. 1 2 x = 75 Simplify. Because EFG is the intercepted arc of D, you need to find mFG in order to find mEFG.

Find the values of x and y. The arc measure of a circle is 360°, so mFG = 360 – 70 – 80 – 90 = 120. y = mEFG Inscribed Angle Theorem 1 2 y = (mEF + mFG) Arc Addition Postulate 1 2 y = (70 + 120) Substitute. 1 2 y = 95 Simplify.

If two inscribed angles of a circle intercept congruent arcs or the same arc, then the angles are congruent. E 1 F 2   H G

A semicircle is 180. What does that make the measure of an arc inscribed on a semicircle? If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle. A P C

Find the values of a and b. An angle inscribed in a semicircle is a right angle, so a = 90. The sum of the measures of the three angles of the triangle inscribed in O is 180. . Therefore, the angle whose intercepted arc has measure b must have measure 180 – 90 – 32, or 58. Because the inscribed angle has half the measure of the intercepted arc, the intercepted arc has twice the measure of the inscribed angle, so b = 2(58) = 116.

If a quadrilateral is inscribed in a circle, then the quadrilateral’s opposite angles are supplementary. X ? ? Isosceles trapezoid

A B   C 2   If a secant (or chord) and a tangent intersect at the point of tangency, then the measure of each angle formed is half the measure of its intercepted arc.

    B C 1 2 A

RS and TU are diameters of A. RB is tangent to A at point R. Find m BRT and m TRS. . m BRT = mRT The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc (Theorem 11-10). 1 2 mRT = mURT – mUR Arc Addition Postulate m BRT = (180 – 126) Substitute 180 for m and 126 for mUR. 1 2 m BRT = 27 Simplify.

Measure is equal to central angle. RS and TU are diameters of A. RB is tangent to A at point R. Find m BRT and m TRS. . Vertical angle Measure is equal to central angle. Measure is equal to ½ of inscribed arc.

Classwork Time!!! Have a great day!

CIRCLES Part 2

Some Vocabulary… Central angle – an angle whose vertex is the center of a circle <APB is a central angle!

Arc – a piece of the outside of a circle Minor arc – measures less than 180° (the measure of its central angle) ; is named by its endpoints Major arc – measures more than 180° (is equal to the difference of 360 and its minor arc) ; is named by its enpoints and a point on the arc

Semicircle – measures exactly 180°; named by its endpoints and a point on the arc

their corresponding chords are congruent. B In a circle or in congruent circles, two minor arcs are congruent if and only if C D their corresponding chords are congruent.   G H E F  

In a circle, if a diameter is perpendicular to a chord, then B E   In a circle, if a diameter is perpendicular to a chord, then the diameter bisects the chord and its arc.

If a chord AB is a perpendicular bisector of another chord, then chord AB is a diameter. C D

they are equidistant from the center. B In a circle or in congruent circles, two chords are congruent if and only if 4 C D P they are equidistant from the center.

they are equidistant from the center. G H In a circle or in congruent circles, two chords are congruent if and only if 2 E F they are equidistant from the center.

Find AB. QS = QR + RS Segment Addition Postulate QS = 7 + 7 Substitute. QS = 14 Simplify. AB = QS Chords that are equidistant from the center of a circle are congruent. AB = 14 Substitute 14 for QS.

P and Q are points on O. The distance from O to PQ is 15 in., and PQ = 16 in. Find the radius of O. . Draw a diagram to represent the situation. The distance from the center of O to PQ is measured along a perpendicular line. . PM = PQ A diameter that is perpendicular to a chord bisects the chord. 1 2 PM = (16) = 8 Substitute. 1 2 OP 2 = PM 2 + OM 2 Use the Pythagorean Theorem. r 2 = 82 + 152 Substitute. r 2 = 289 Simplify. r = 17 Find the square root of each side. The radius of O is 17 in. .

Classwork Time!!! Have a great day!

I J F

Two tangent segments that intersect at the same exterior point are congruent.

 

AB (AB+BC) = AD (AD+DE) AB  AC = AD  AE AB (AB+BC) = AD (AD+DE) Outside Times Whole = Outside Times Whole External Times Secant Segment = External Times Secant Segment

UV  UV = UX  XY UV  UV = UX  (UX + XY) Outside Times Whole = Outside Times Whole With the tangent segment, the outside and the whole thing are the same!

CIRCLES Part 3

Arc Length A 60° B P C Arc length: The length of an arc is different from the measure of an arc. If a circle were made of string, the arc length would be the length of the piece of string representing the arc.

60° Find what part of the circle is represented by the central angle: Arc Length   Find what part of the circle is represented by the central angle: A 60°   B P C IF PB = 9, then the circumference = ___________. Multiply the circumference by The length of the arc = _________.

Find the length of ADB in M in terms of . . Because mAB = 150, mADB = 360 – 150 = 210. Arc Addition Postulate mADB 360 length of ADB = • 2 r Arc Length Formula length of ADB = • 2 (18) Substitute. 210 360 length of ADB = 21 The length of ADB is 21 cm.

Lesson Quiz 1. A circle graph has a section marked “Potatoes: 28%.” What is the measure of the central angle of this section? 2. Explain how a major arc differs from a minor arc. Use O for Exercises 3–6. 3. Find mYW. 4. Find mWXS. 5. Suppose that P has a diameter 2 in. greater than the diameter of O. How much greater is its circumference? Leave your answer in terms of . 6. Find the length of XY. Leave your answer in terms of . . 100.8 A major arc is greater than a semicircle. A minor arc is smaller than a semicircle. 30° 270° 2 9

Classwork Time!!! Have a great day!