Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central

Circles

Preliminaries

•A • B•P

•C

_AB is a semicircle_AC is a minor arc_

BAC is a major arc

Preliminaries

•A • B•P

•C

_AB is a

semicircle_AC is a minor arc_

BAC is a major arc

Preliminaries

•A • B•P

•C

_AB is a semicircle_AC is a

minor arc_

BAC is a major arc

Preliminaries

•A • B•P

•C


BAC is a

major arc

Preliminaries

•A • B•P

•C


BAC is a major arc

Preliminaries

•A

•B

•P

m◦

Definition∠APB is called a central angle.

Relationship between m∠APB and m_AB?

Preliminaries

•A

•B

•P

m◦

Definition∠APB is called a central angle.

Relationship between m∠APB and m_AB?

Preliminaries

Central AnglesThe measure of a minor arc is the same as the central angle cutting offthat arc.

Central Angles

The measure of a major arc is 360◦ less the measure of the centralangle cutting off the associated minor arc.

DefinitionThe minor arc is also called the intercepted arc.

Preliminaries


Central Angles



Preliminaries


Central Angles



Arc Addition Postulate

•A

•B

•P

•C

Arc Addition Postulate

If C is on_AB, then

m_AB = m

_AC + m

_CB

Arc Addition

•A •B

•P

•C

•D

Postulate

If C and D are on_AB and m

_AD = m

_BC, then

m_AC = m

_BD

Concentric Circles

•A •B

•P

•X

•Y

120◦

m_AB? 120◦

m_XY? 120◦

Are the arcs congruent?

What we can sayCongruent central angles have the same intercepted arcs andcongruent arcs have the same measured central angles.

Concentric Circles

•A •B

•P

•X

•Y

120◦

m_AB? 120◦

m_XY? 120◦



Concentric Circles

•A •B

•P

•X

•Y

120◦

m_AB?

120◦

m_XY? 120◦



Concentric Circles

•A •B

•P

•X

•Y

120◦

m_AB? 120◦

m_XY? 120◦



Concentric Circles

•A •B

•P

•X

•Y

120◦

m_AB? 120◦

m_XY?

120◦



Concentric Circles

•A •B

•P

•X

•Y

120◦

m_AB? 120◦

m_XY? 120◦



Concentric Circles

•A •B

•P

•X

•Y

120◦

m_AB? 120◦

m_XY? 120◦



Concentric Circles

•A •B

•P

•X

•Y

120◦

m_AB? 120◦

m_XY? 120◦



Midpoints

•A •B

•P

•C

DefinitionThe midpoint of an arc divides the curve into two congruent curves.

Midpoints

•A •B

•P

•C

DefinitionThe midpoint of an arc divides the curve into two congruent curves.

Midpoints

Example

B is the midpoint of_AC. Prove ∆APB ∼= ∆CPB.

•A •C

•P

•B

∠APB ∼= ∠CPB

BP ∼= BP

AP ∼= CP

SAS

Midpoints

Example


•A •C

•P

•B

∠APB ∼= ∠CPB

BP ∼= BP

AP ∼= CP

SAS

Midpoints

Example


•A •C

•P

•B

∠APB ∼= ∠CPB

BP ∼= BP

AP ∼= CP

SAS

Midpoints

Example


•A •C

•P

•B

∠APB ∼= ∠CPB

BP ∼= BP

AP ∼= CP

SAS

Midpoints

Example


•A •C

•P

•B

∠APB ∼= ∠CPB

BP ∼= BP

AP ∼= CP

SAS

Midpoints

Example


•A •C

•P

•B

∠APB ∼= ∠CPB

BP ∼= BP

AP ∼= CP

SAS

Midpoints

Example


•A •C

•P

•B

∠APB ∼= ∠CPB

BP ∼= BP

AP ∼= CP

SAS

Related Statement

•A •C

•P

•B

TheoremIn the same or congruent circles, congruent arcs have congruentchords.

Related Statement

•A •C

•P

•B

TheoremIn the same or congruent circles, congruent arcs have congruentchords.

An Example

Example

Given that CA ∼= DB, prove ∠CAD ∼= ∠DBC.

•P

•A •B

•C •D

Diameter and Chords

TheoremIn a circle, a diameter drawn perpendicular to a chord bisects thechord and its arc.

•P

•A •B

Diameter and Chords

TheoremIn a circle, a diameter drawn perpendicular to a chord bisects thechord and its arc.

•P

•A •B

An Example

ExampleThe length of the diameter of a circle is 20 and the length of the chordAB is 16. What is the shortest distance between the chord and thecenter of the circle?

•P

•A •B

d

8

10d = 6

An Example


•P

•A •B

d

8

10d = 6

An Example


•P

•A •B

d

8

10d = 6

An Example


•P

•A •B

d

8

10

d = 6

An Example


•P

•A •B

d

8

10d = 6

Parallel Lines

Parallel LinesIn a circle, parallel lines cut off equal arcs.

Example

Given AB||CD, prove_AC ∼=

_BD

• P

•A •B•C

•D•

M

Think ‘difference of arcs’ ...

Parallel Lines


Example


_BD

• P

•A •B•C

•D

•M


Parallel Lines


Example


_BD

• P

•A •B•C

•D•

M


Parallel Lines


Example


_BD

• P

•A •B•C

•D•

M


More Parallel Lines

Example

If CD||AB and AB is the diameter of the circle, find the measure of_AC

and_

BD if m_

CD = 40◦.

• P•A • B

•C

•D

Both arcs are 70◦.

More Parallel Lines

Example


and_

BD if m_

CD = 40◦.

• P•A • B

•C

•D


More Parallel Lines

Example


and_

BD if m_

CD = 40◦.

• P•A • B

•C

•D


Tangents and Secants

DefinitionTangent lines intersect a circle at exactly one point.

DefinitionSecant lines intersect a circle at exactly two points.

•P•A

•C •D




•P•A

•C •D




•P•A

•C •D

You know you love proofs

TheoremA radius drawn to a point of tangency is perpendicular to the tangent.

• P•A

B •

•X



• P•A

B •

•X



• P•A

B •

•X

The Proof

Proof.By contradictionSuppose PA 6⊥ ←−AB. Then there must be another segment, say PX, suchthat PX ⊥ ←−AB

Since PX ⊥ ←−AB, it must be the shortest distance to AB,But, X is exterior to the circle, so mPA < mPX, a contradiction.Therefore, PA ⊥ AB.

The Proof

Proof.By contradictionSuppose PA 6⊥ ←−AB. Then there must be another segment, say PX, suchthat PX ⊥ ←−ABSince PX ⊥ ←−AB, it must be the shortest distance to AB,

But, X is exterior to the circle, so mPA < mPX, a contradiction.Therefore, PA ⊥ AB.

The Proof

Proof.By contradictionSuppose PA 6⊥ ←−AB. Then there must be another segment, say PX, suchthat PX ⊥ ←−ABSince PX ⊥ ←−AB, it must be the shortest distance to AB,But, X is exterior to the circle, so mPA < mPX, a contradiction.

Therefore, PA ⊥ AB.

The Proof

Proof.By contradictionSuppose PA 6⊥ ←−AB. Then there must be another segment, say PX, suchthat PX ⊥ ←−ABSince PX ⊥ ←−AB, it must be the shortest distance to AB,But, X is exterior to the circle, so mPA < mPX, a contradiction.Therefore, PA ⊥ AB.

Tangent Segment

DefinitionA tangent segment is a line segment that has a point on the tangentline and the point of tangency as its endpoints.

• P•A

B •

Tangent Segment

DefinitionA tangent segment is a line segment that has a point on the tangentline and the point of tangency as its endpoints.

• P•A

B •

Tangent Segment

ExampleThe length of a tangent segment drawn from point B to a circle P is 12units. If the radius of circle P is 5 units, find the distance from thepoint B to the center of the circle.

• P•A

B •

12

5

d = 13

Tangent Segment


• P•A

B •

12

5

d = 13

Tangent Segment


• P•A

B •

12

5

d = 13

Tangent Segment


• P•A

B •

12

5

d = 13

Inscribed and Circumscribed Polygons

DefinitionA polygon is circumscribed about a circle if each of its sides aretangent to the circle.

DefinitionA polygon is inscribed in a circle if each of its vertices are on thecircle.

ExampleA quadrilateral is inscribed in a circle such that the sides divide thecircle into arcs whose measures have the ratio 1 : 2 : 3 : 4. how manydegrees is each arc?

36◦, 72◦, 108◦, 144◦





36◦, 72◦, 108◦, 144◦





36◦, 72◦, 108◦, 144◦





36◦, 72◦, 108◦, 144◦

Angle Measurement: Vertex on the Circle

DefinitionAn inscribed angle has it’s vertex on the circumference of the circleand it’s sides are secants.

• P

x◦

2x◦

Angle Measure

The measure of an inscribed angle is equal to 12 the measure of its

intercepted arc.



• P

x◦

2x◦

Angle Measure


intercepted arc.



• P

x◦

2x◦

Angle Measure


intercepted arc.

Examples

ExampleFind x.

•

x◦

••

•

110◦

70◦

Examples

ExampleFind x.

•

x◦

••

•

110◦70◦

Examples

ExampleA triangle is inscribed in a circle so that its sides divide the circle intoarcs whose measures have the ratio 2 : 3 : 7. Find the measure of thelargest angle of the triangle.

•

•

•

•

105◦

Examples


•

•

•

•

105◦

Examples


•

•

•

•

105◦

Examples

Example

Quadrilateral ABCD is inscribed in a circle. If m∠A = x◦ andm∠B = y◦, find angles C and D in terms of x and y.

•

•

•

• •

A

B

DC

x◦y◦

m∠C = (180− x)◦, m∠D = (180− y)◦.

TheoremOpposite angles of an inscribed quadrilateral are supplementary.

Examples

Example


•

•

•

• •

A

B

DC

x◦y◦

m∠C = (180− x)◦, m∠D = (180− y)◦.


Examples

Example


•

•

•

• •

A

B

DC

x◦y◦

m∠C = (180− x)◦, m∠D = (180− y)◦.


Examples

Example


•

•

•

• •

A

B

DC

x◦y◦

m∠C = (180− x)◦, m∠D = (180− y)◦.


More Tangency

TheoremThe measure of an angle formed by a tangent and a chord drawn tothe point of tangency is equal to half the measure of the interceptedarc.

• P

•A • Bx◦

2x◦

•C

More Tangency

TheoremThe measure of an angle formed by a tangent and a chord drawn tothe point of tangency is equal to half the measure of the interceptedarc.

• P

•A • Bx◦

2x◦

•C

More Tangency

Example

Given that BC is tangent to circle P at point B, prove the measure of

angle x is half of the measure of_AB.

• P

•A • Bx◦

•C

•D

AD||BC

More Tangency

Example



• P

•A • Bx◦

•C

•D

AD||BC

More Tangency

Example



• P

•A • Bx◦

•C

•D

AD||BC

Examples

Example

Find the value of x if m_

ACB = 250◦

• P

•A • C

•B

x◦

m_

ACB = 250◦

m_AB = 110◦

x =12

(110◦) = 55◦

Examples

Example


ACB = 250◦

• P

•A • C

•B

x◦

m_

ACB = 250◦

m_AB = 110◦

x =12

(110◦) = 55◦

Examples

Example


ACB = 250◦

• P

•A • C

•B

x◦

m_

ACB = 250◦

m_AB = 110◦

x =12

(110◦) = 55◦

Examples

Example


ACB = 250◦

• P

•A • C

•B

x◦

m_

ACB = 250◦

m_AB = 110◦

x =12

(110◦) = 55◦

Examples

Example


ACB = 250◦

• P

•A • C

•B

x◦

m_

ACB = 250◦

m_AB = 110◦

x =12

(110◦) = 55◦

Example

Example

Find the measure of the numbered angles if m_AC = 110◦.

• P ••

•

A B

C1 2

3 4 5

6

Example

m∠1 = 12(110) = 55◦

m_

BC = 180− 110 = 70◦ ⇒ m∠2 = 12(70) = 35◦

m∠3 = 35◦

m∠4 = 110◦

m∠5 = 55◦

m∠6 = 12(

_AB) = 1

2(180) = 90◦

Chord-Chord-Angle Theorem

TheoremThe measure of an angle formed by two chords intersecting in theinterior of a circle is equal to 1

2 the sum of the intercepted arcs.

• P•

•

A

B•

•C

D

Example

ExampleFind the value of x.

• P•

•

A

B•

•C

Dx

65◦105◦

x = 85◦

Example


• P•

•

A

B•

•C

Dx

65◦105◦

x = 85◦

Example


• P•

•

A

B•

•C

D108◦62◦

x

x = 154◦

Example


• P•

•

A

B•

•C

D108◦62◦

x

x = 154◦

Activity; The Magic Circle

Materials:

1 one 7 inch circle2 pencil3 ruler (optional)

The reason that the ruler is optional is because you can, at any time,include the measurement of area, perimeter, surface area, or volume.


Materials:

1 one 7 inch circle2 pencil3 ruler (optional)

The reason that the ruler is optional is because you can, at any time,include the measurement of area, perimeter, surface area, or volume.


Step 1Look at the shape you are holding. What is it?

circle

What is the distance around the outside called? circumference


Step 1Look at the shape you are holding. What is it? circle




What is the distance around the outside called?

circumference





Step 2Fold your circle directly in half and crease it well.

Open the circle. What is this crease? diameter



Open the circle. What is this crease?

diameter



Open the circle. What is this crease? diameter


Step 3Hold the circle at the ends of the crease. Fold your circle in half again,but this time match up the end points of the crease.

Open your circle, is this also a diameter? How do you know?

Is there something special about the way these lines intersect?The lines are perpendicular








Is there something special about the way these lines intersect?

The lines are perpendicular






Step 4Place a dot, no bigger than the width of a pencil, at the point wherethe creases connect.

What is this point called? center



What is this point called?

center



What is this point called? center


Step 5Using your pencil, trace one of the lines from the center to the edge ofthe circle.

What is this line called? radius



What is this line called?

radius



What is this line called? radius


Step 6Fold in one of the outer, curved edges of the circle until it just touchesthe dot in the middle. Crease it well.

Open the fold and look at the crease you just made. Is it a diameter?Is it a radius? Why or why not?

What is this segment called? chord








What is this segment called?

chord






Step 7Look at the curved part of the circle between the points where thisline touches the outside of the circle.

What is this curved part called? arc



What is this curved part called?

arc



What is this curved part called? arc


Step 8Take the opposite side of your circle and fold it so that the curved partjust touches the center and the bottom forms a perfect point.

What does this look like? cone

Crease this well.



What does this look like?

cone

Crease this well.




Crease this well.




Crease this well.


Step 9Fold the top of your ice cream cone down until the curved part justtouches the center of the circle. The top corners should make perfectpoints, crease well. Now describe the shape you have.

Do you notice anything special about this triangle?It is an equilateral triangle.



Do you notice anything special about this triangle?

It is an equilateral triangle.



Do you notice anything special about this triangle?It is an equilateral triangle.


Step 10Fold the new triangle in half by matching up two of the points. Creasewell. The new crease splits the triangle in half. What is this linecalled?

altitude

Do you notice anything else about this triangle?It is a right triangle


Step 10Fold the new triangle in half by matching up two of the points. Creasewell. The new crease splits the triangle in half. What is this linecalled? altitude




Do you notice anything else about this triangle?

It is a right triangle





Step 11Open the right triangle up to the equilateral triangle.

Take the top corner of the big triangle and fold it along the crease ofthe height. You can match the top point up to the bottom crease line.On the inside you will now see three smaller triangles.

Turn the paper over so that you do not see the creases. What is thisshape called? trapezoid








Turn the paper over so that you do not see the creases. What is thisshape called?

trapezoid






Step 12Turn it back over so that you now see all of the creases. Fold one ofthe outer triangles in so that it lies directly on top of the centertriangle.

Turn it back over. What shape do yo see? rhombus



Turn it back over. What shape do yo see?

rhombus



Turn it back over. What shape do yo see? rhombus


Step 13Turn your shape back over and fold the last outer triangle over ontothe center one again. You should now have a smaller equilateraltriangle.


Step 14Open up all three of the small triangles. Bring the three loose pointstogether.

What shape do you have now? pyramid

Note: this would be an opportunity to discuss bases, faces,verticesand edges.



What shape do you have now?

pyramid











Step 15Open your pyramid back up to the large equilateral triangle.

Fold over one of the points so that it just touches the dot in themiddle. What shape have you re-created?


Step 15Open your pyramid back up to the large equilateral triangle.

Fold over one of the points so that it just touches the dot in themiddle. What shape have you re-created?


Step 16Fold one more of the points in so that it just touches the dot in themiddle. Now what shape do you have?

pentagon

Now fold in the last point. What shape is it now? hexagon

Note: this is a good place to discuss planar figures


Step 16Fold one more of the points in so that it just touches the dot in themiddle. Now what shape do you have? pentagon





Now fold in the last point. What shape is it now?

hexagon











Step 17Turn to the other side and fit one of the corners into a flap on theopposite side of the triangle. You may have to try more than one.Choose the one that makes the best fit. Slide the last cornerunder/inside the others.

What shape do you have now? truncated tetrahedron



What shape do you have now?

truncated tetrahedron



What shape do you have now? truncated tetrahedron

Documents

Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central