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CIRCLE GEOMETRY
Chapter 8
8.1 – PROPERTIES OF
TANGENTS TO A CIRCLE
Chapter 8
DEFINITIONS
A tangent line is a line that intersects a circle at only one point.
The point where the tangent intersects the circle is the point of tangency.
CIRCLES AND TANGENTS
A tangent to a circle is perpendicular to the radius at the point of
tangency. That means that ∠APO = ∠BPO = 90º.
EXAMPLE
Point O is the centre of a circle and AB is tangent to the
circle. In ΔOAB, ∠AOB = 63º. Determine the measure of
∠OBA.
Which angle are we looking for?
90º
AB is tangent to the circle. What does that
mean about ∠OAB?
 ∠OAB = 90º
What do the angles in a triangle add up to?
The angles of a triangle always sum to 180º.
 63º + 90º + ∠OBA = 180º
 ∠OBA = 180º – 90º – 63º
 ∠OBA = 27º
∠OBA = 27º
EXAMPLE
Point O is the centre of a circle and CD is a tangent
to the circle. CD = 15 cm and OD = 20 cm.
Determine the length of the radius OC to the
nearest tenth.
What can we say about ∠OCD?
Since CD is tangent to the circle, ∠OCD = 90º.
What theorem can we use for right angle triangles?
The Pythagorean Theorem: a2 + b2 = c2
 a2 + 152 = 202
 a2 = 202 – 152
 a2 = 400 – 225
 a2 = 175
 a = 13.2 cm
The radius is 13.2 cm.
EXAMPLE
An airplane, A, is cruising at an altitude of 9000 m.
A cross section of Earth is a circle with radius
approximately 6400 km. A passenger wonders how far she
is from point H on the horizon she sees outside the window.
Calculate this distance to the nearest kilometre.
6400 km
What is the length of the third side of the triangle?
It’s the radius. Are we given the radius anywhere else
in the diagram?
 The radius is constant anywhere in the circle.
a2 + b2 = c2
a=?
b = 6400 km
c = 6400 + 9
= 6409 km
 a2 + 64002 = 64092
 a2 = 64092 – 64002
 a2 = 115281
 a = 339.53
What is 9000 m in km?
 9000 m = 9 km
The distance to point
H is 340 km.
PG. 388–391, #5, 6, 7,
9, 13, 17, 20
Independent
Practice
8.2 – PROPERTIES OF
CHORDS IN A CIRCLE
Chapter 8
CHORDS
Follow the steps outlined on page 392.
DEFINITIONS
A chord is a line segment that joins two points on a circle.
The diameter of a circle is a chord that goes through the centre of the circle.
Properties of Chords:
The perpendicular
from the centre of
a circle to a chord
bisects the chord.
The perpendicular
bisector of a chord
in a circle passes
through the centre
of the circle.
A line that joins the centre of
a circle and the midpoint of a
chord is perpendicular to the
chord.
EXAMPLE
Point O is the centre of a circle, and line segment OC
bisects chord AB.
∠OAC = 33º
Determine the values of xº and yº.
90º
What type of triangle is ΔOAB?
It’s an isosceles triangle, because it has two equal sides.
 And that means it also has two equal angles. So what
is xº?
 xº = 33º
Since OC bisects chord AB, what can we say about ∠OCA?
OC must be perpendicular to AB, so ∠OCA must be 90º.
 33º + 90º + yº = 180º
xº = 33º, yº = 57º
 yº = 180º - 90º - 33º
 yº = 57º
EXAMPLE
Point O is the centre of a circle.
AB is a diameter with length 26 cm.
CD is a chord that is 10 cm from the centre of the circle.
What is the length of chord CD, to the nearest tenth?
13 cm
What’s the radius of the circle?
r = 13 cm
What’s the length of OC?
It’s from the centre to a point on the circle, so it’s the radius of the circle.
 OC = 13 cm
a2 + b2 = c2
 a2 + 102 = 132
 a2 = 132 – 102
 a2 = 169 – 100
 a2 = 69
 a = 8.307
So, if CE is 8.307, what’s CD?
CD = 8.307 X 2 = 16.6 cm
TRY IT
PG. 397-399, #4, 5, 7,
10, 11, 12, 14.
Independent
Practice
8.3 – PROPERTIES OF
ANGLES IN A CIRCLE
Chapter 8
ANGLES IN A CIRCLE
Follow the steps outlined on page 404-405.
DEFINITIONS
An arc is a section of the circumference (the outside) of a circle.
 The shorter arc AB is the minor arc.
 The longer arc AB is the major arc.
A central angle is the angle formed by joining the
endpoints of an arc to the centre of the circle.
An inscribed angle is the angle formed by joining the
endpoints of an arc to a point on the circle.
We say that the inscribed and central angles in this circle
are subtended by the minor arc AB.
CENTRAL AND INSCRIBED ANGLES PROPERTIES
In a circle, the measure of a central angle subtended by
an arc is twice the measure of an inscribed angle
subtended by the same arc.
∠POQ = 2∠PRQ
In a circle, all inscribed angles subtended by the same
arc are congruent.
 ∠PTW = ∠PSQ = ∠PRQ
CENTRAL AND INSCRIBED ANGLES PROPERTIES
All inscribed angles subtended by a semicircle are
right angles (90º).
EXAMPLE
Determine the values of xº and yº.
Which angles are central angles and which are
inscribed angles?
∠ACB and ∠ADB are inscribed angles, subtended by the
same arc AB, so they must be equal.
 ∠ACB = 55º
 ∠ADB = 55º
∠AOB is a central angle.
 Is ∠AOB going to be half of 55º or twice 55º?
Central angles are double the inscribed angles.
 ∠AOB = 110º
xº = 55º
yº = 110º
EXAMPLE
Rectangle ABCD has its vertices on a circle with radius 8.5 cm.
The width of the rectangle is 10.0 cm. What is its length, to the
nearest tenth of a centimetre?
The angles of the rectangle are all 90º.
 ∠ABC = ∠ADC = 90º.
What can we say about AC if its inscribed angles are 90º?
 AC is the diameter
 AC = 8.5 x 2 = 17 cm
a2 + b2 = c2
 a2 + 102 = 172
 a2 = 172 – 102
 a2 = 189
 a = 13.7
The rectangle is
13.7 cm long.
EXAMPLE
Triangle ABC is inscribed in a circle, centre O.
∠AOB = 100º and ∠COB = 140º
Determine the values of xº, yº, and zº.
What’s the angle of a full circle?
 360º. So, the angle all the way around the origin needs
to add up to 360º.
 100º + 140º + xº = 360º
 xº = 120º
yº is an inscribed angle. What’s the central angle subtended by the same arc?
xº is the central angle subtended by the same arc as yº. Will yº be half of xº or
double xº?
 yº = 120º/2 = 60º
How might we find angle zº? What type of triangle is AOC?
PG. 410-412, #3, 4, 5, 6,
9, 11, 13, 15
Independent
Practice
CHALLENGE
What is the measure of yº?
EXAM QUESTION EXAMPLE
A line that intersects a circle at only one point.
tangent
The shorter section of the circumference between
two points on a circle.
Minor arc
The distance from the centre of a circle to any
point on its circumference.
Radius
A line segment that joins two points on a circle.
chord
A chord that passes through the centre of a circle.
diameter
The angle formed by joining the endpoints of an
arc to the centre of the circle.
Central angle
The point where the tangent intersects the circle.
Point of tangency
The larger section of the circumference between
two points on a circle.
Major arc
The distance around a circle.
circumference
The angle formed by joining the endpoints of an
arc to a point on the circle.
Inscribed angle