Chapter 8CIRCLE GEOMETRY
Chapter 8
8.1 – PROPERTIES OF TANGENTS TO A
CIRCLE
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?
AB is tangent to the circle. What does that mean about ∠OAB?
∠OAB = 90º
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. 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.
6400 km
a2 + b2 = c2
a = ?b = 6400 kmc = 6400 + 9 = 6409 km
a2 + 64002 = 64092
a2 = 64092 – 64002
a2 = 115281 a = 339.53
The distance to point H is 340 km.
What is 9000 m in km? 9000 m = 9 km
Independent Practice
PG. 388–391, #5, 6, 7, 9, 13, 17, 20
Chapter 8
8.2 – PROPERTIES OF CHORDS IN A CIRCLE
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.
Since OC bisects chord AB, what can we say about ∠OCA?OC must be perpendicular to AB, so ∠OCA must be 90º.
90º
33º + 90º + yº = 180º
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
yº = 180º - 90º - 33º
yº = 57º
xº = 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?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
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
Independent Practice
PG. 397-399, #4, 5, 7, 10, 11, 12, 14.
Chapter 88.3 – PROPERTIES OF ANGLES IN A CIRCLE
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 = 55ºyº = 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 = 189a = 13.7
The rectangle is 13.7 cm long.
EXAMPLE
Triangle ABC is inscribed in a circle, centre O.∠AOB = 100º and ∠COB = 140ºDetermine 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?
Independent Practice
PG. 410-412, #3, 4, 5, 6, 9, 11, 13, 15
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