Chapter 8CIRCLE GEOMETRY

Chapter 88.1 – 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