Space and ShapeGrade 9 Math

Discovering Circle Theorems

E6

Recognize, name, describe and represent: arcs, chords, tangents, central angles, inscribed angles and

circumscribed angles. Make generalizations about their relationships in circles.

To review for this lesson…

• Circumference: the distance outside of a circle

• Diameter: the distance across the longest part of a circle

• Radius: the distance from the center to the outside of a circle

• Segment: essentially a “chunk” of circle, made of 2 radii and an arc

• Centre: the most central point in a circle, equally distant from all other parts

• Chord: a line that crosses the circle, but does not divide into equal parts (not the diameter)

Parts of the Circle

CircumferenceDiameter

Chord Radius

Sector Center

Tangent Arc

x

a b

c

d

1. Mark two points a and b on the circumference.

2. Draw lines from a and b to another point on the circumference. Call this ‘c’

3. Draw lines from a and b to the centre. Call this ‘d’. 

4. Measure angle acb. Measure angle adb

What can you observe about the angle at the centre and angle at the circumference?

Activity #1

x

x

x

1. Draw the diameter of the circle to form 2 semi-circles 2. In one half, mark any point (x) on the circumference and draw lines from each end of the diameter to that point 3. Measure the angle you have created.

4. Repeat the process in the second half of your circleNow we have two triangles. What can you say about the

triangle formed in a semi-circle?

Activity #2

ab

c

e f

1. Mark three points a, b and c anywhere in the top half of the circumference and then points e and f in the bottom half, again on the circumference 2. Draw 2 straight lines each from a, b, and c to points e and f. (six lines in total) 3. Measure the three angles eaf, ebf and ecf.

What can you say about the relationship between the three angles?

Activity #3

1. Draw a quadrilateral where all four vertices are on the circumference of the circle.

2. Measure the angle of each vertex in your quadrilateral.

x

x

x

x

What do you notice about opposite angles in your cyclic quadrilateral?

Activity #4

x

1. Draw the radius on your circle.

2. Draw a tangent which touches the circumference where the radius does.

3. Measure the angle between the radius and the tangent.

4. Repeat the process again at a different point on the circumference

What do you notice about the angle created?

Activity #5

1. Draw two tangents to the circle which cross at a point outside of the circle.

2. Measure the distance from the where each tangent meets the circle to the point.

What do you notice about the lengths?

Activity #6

To clarify….

• https://www.youtube.com/watch?v=WufDdk1MZTg

_____________: an angle made by…

The angle subtended at the centre of a circle (by an arc or chord) _____________by the same arc or chord. (angle at centre)

2xo

2xo 2xo 2xo

2xo 2xo

Theorem 1

xo

xo

xoxo

xo xo

o oo o

o o

Watch for this one later.

o Diameter

The angle in a _____________is a right angle

Or, the angle _____________to the circumference is a right angle

Theorem 2

This is just a special case of Theorem 1 and is referred to as a theorem for convenience.

Th2

Angles subtended by an arc or chord in the same _____________Theorem

3

xo xo

xo

xo

xo

yo

yo

Th3

Cyclic Quadrilateral TheoremTheorem 4

The _____________of a cyclic quadrilateral are supplementary. (They sum to 180o)

w

x

y

zq

p

r

s

Th6

The angle between a tangent and a _____________ (Tan/rad)Theorem

5

o

Two Tangent TheoremTheorem 6

From any point outside a circle _____________ can be drawn and they are equal in length.

P

T

UQ

R

PT = PQ

P

T

U

Q

R

PT = PQ

Th7

The Alternate Segment TheoremTheorem 7The angle between a _____________ through the point

of contact is equal to the angle subtended by that chord in the alternate segment.

xo

xo

yo

yo

45o

xo

yo

60o

zo

**Find the missing angles below giving reasons in each case.

Th5