CIRCLES

Definition of Terms

Circle, Chord, Radius, Diameter, Circumference, Arc, Sector, Centre and Segment of a Circle Define circle, chord. radius, diameter, circumference, arc, sector. centre and segment ofa circle A cirele: is the locus or the set ofall points equidistant from a fixed point ealled the center.

Arc: a curved line that is part ofthe circumference of a circle Chord: a line segment within a cirele that touches 2 points on the circle Circumference: The distance around the circle Diameter: The longest distance from one end of a circle to the other Origin: the center ofthe circle Pi(n):A number, 3.141592… equal to (the circumference) / (the diameter) of any circle Radius: distance fiom center of circle to any point on it

Scetor: is like a slice of pic (a circle wedge). ‘Tangent of circle: line perpendicular to the radius that touches ONLY one point on the circle.

Central Angle

‘The Formula for the Length of an Arc Derive the formula for the length of an are

‘The Central Angle

Caleulate the central angle A central angle is an angle Formed by two intersecting radii such that its vertex is at thecenter of the circle as 8 A 80°

‘The Concept of Radian Measure Explain the concept of radian measure Radians are the standard mathematical way to measure angles. One radian is equal to the angle created by taking the radius ofa circle and stretching it along the edge of the etcle The radian is a pure mathematical measurement and therefore is preferred by mathematicians ‘over degree measures. For use in everyday work, the degree is easier to work with, but for purely

mathematical pursuits, the radian gives better results You probably will never see radian measures used in construction or surveying, but it is a common unit in mathematies and physies.

ca Radians to Degree and Vice Versa Convert radians to degree and vice versa ‘The unit used to describe the measurement of an angle that is most familiar is thedegree, To convert radians to degrees or degrees to radians, the following relationship can be used.

Example 1

Convert 45 degrees to radians

Solution

radians ~ 45/5732

Most often when writing degree measure in radians, pi is not calculated in, so for this problem,

Example 2

Conver pi/3 radians to degree

Angles Properties

Circle Theorems of Inscribed Angles

ig Sc i wn i ei a Nal so be othe eel: arte antiga Cyn, waxes lair betwee inscribed angles and central angles having the same subtended arc. The angle of the subtended P bad Fi R In the first circle,is a central angle subtended by are. Angleis an inscribed angle subtended by sta. i aa nel eh ml Swe Hien ig ib 9 a a RN i i i aw sir sees At our oben tarsal sc wea ar Bal mit i i ass Theorem The measure of an angle inscribed in a circle is half the measure of the arc it intercepts. Note that

this is equivalent to the measure of the inscribed angle is half the measure of the central angle if they intercept the same are Proof Letbe an inscribed angle andbe a central angle both subtending areas shown in the figure. Draw line. This forms two isosceles trianglesandsince two of their sides are radi ofthe circle Ss (7 iv In triangle, if we let the measure ofhe, then angleis also. By theexterior angle theorem, the

measure of angle. This is also similar to triangle. If we let angle, it follows thatis equall 10 2y. In effect, the measure of the inseribed angleand the measure of central anglewhich is what we want to prove.

The Circle Theorems in Solving Related Problems

Apply the circle theorems in solving related problems

Example 3

‘An are subtends an angle of 200 at the center of the circle of radius2Sem.Find the length of this

Solution

nro

3.142%25%20 al a SE 8.73em The length ofthe are is 8.73cm,

Example 4

Anare of lengih Sem sublends 0° at the cenerof the circle, whit is theradius ofthe circle?

, aro 180° _ 180°xSem

• The radius is 5.7cm.

Chord Properties of'a Circle

Chord Properties of a Circle Identify chord properties ofa circle us Imagine that you are on one side of @ perfectly circular lake and looking across to a fishing pier on the other side. The chord is the line going across the citele from point A (you) to point B (the fishing pier). The circle outlining the lake's perimeter is called thecircumference Achord of @ cireleis a line that connects two points on a circle's circumference.

To illustrate further, let's look at several points of reference on the same cireular lake from before. IF each point of reference (1c. duck feeding area, picnic tables, you, water fountain, and fishing pier) were directly on this lake's circumference, then each line connecting a point to another point on the circle would be chords.

Water Fountain

Fishing Pier B

A You c

Duck Fee

E 0 ‘icnic Tables

• The line between the fishing pier and you is now chord AC
• The line between the water fountain and duck feeding area is now chord BE
• The Tine between you and the picnic tables is chord CD

If we had a chord thet went directly through the center of a eitele it would be called adiameter. It we had a line that did not stop at the circle's circumference and instead extended into infinity, it would no longer be a chord; it would be called aseeant, The Theorem on the Perpendicular Bisector to a Chord Prove the theorem an the perpendicular bisector (a a chord.

Proof of Theorem

A is, . B D Given: ©O, OD 1 AB

Prove: OD bisects AB

a a aw GA GB Hiwo points determine exacly one f. [Draw A, OB b neg B. <O8A <O89 are nahe rales , Pepenlialar is eet to Tom Hig ce angles.

• AOBA, AOEB arc right triangles 4, A Fight tangle contains one right
• aazoe 5, Redii in a circle are congruent

—— [Reflexive Property -A segments

• OE=OE [congruent to itself.

[HL Ifthe hypotenuse and leg of onc whe light triangle are congrucat tothe fe AAOR a ABs: – Jeorresponding parts of another triangle, the triangles are congruent ICPCTC – Corresponding parts of b. [zene fs ere cnereeeeee.

[Midpoint of a line segment isthe

• E is the midpoint of AE '9. point on that line segment that divides

the segment two congruent segments. [Bisector of a line segment is any line 10.) GD bisects AB 10. (or subset of a Line) that intersects the Jreament ac its midpoint, ‘The Theorem on Parallel Chords Prove the theorem on parallel chords ww Parallel chords in the same circle always cut congruent arcs. Parallel chords intercept congruent

• Const inter perpen toh pall od
• Ret sss the ameter (done dams) What appens othe endo? The

‘The Theorems on Chords in Solving Related Problems

Example 5

fo a aS 20 we) KAY

i sacar

a ofl \ s\ J igs B— ePN of [No \ by va Vf / ™

Kup J

aie re

z

On} = OP +RP*

soap Ret

= RP=VE=16 = 3m

Tangent Properties

A Tangent to a Cirele Tangent is a line which touches a circle The point where the line touches the citcle is called the point of contact. A tangent is pependicular to the ridius atthe point of contact.

‘Tangent Properties of a Circle Identity tangent properties ofa circle 4 line that 18a tangent to each of two circles. A common extemal tangent does not imtersect the ‘Tangent ‘Theorems Theorem 1

Ke L \

S \ WY i

Intersecting Chords Rule: (segment picce)»(segment piece) (segment picce)»(segment piece)

Theorem Proof

D (NX) fess

Nhat Ye

ne Given: Chords 4B and CD

Prove: AEEB =CE+ED

ut [Statements] Reasons [F]eiis Ait (Re Biel 2 ive ott dseenins only on ie. langles of another triangle, the oe S. AZ BD 5. {Corresponding sides of similar ce” eB ltiangles are in proportion.

lmeans equals the product of the Theorem 2: If two secant segments are drawn to a circle from the same extemal point, the product of the Em

La ‘ Theorem 3: product of the length of the sccant segment and its external part equals the square ofthe length of a Cy me

‘Theorems Relating to Tangent to a Circle in Solving Problems Apply theorems relating 1 tangent toa cirele in solving problems

Example 7

angle formed between the tangents.

Solution

‘Two tangents and two radii form a figure with 360% If y is the angle formed between the

The angle formed between tangents is 40 degrees Pra