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.
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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.
A tangent to a circle is perpendicular to the radius at the point of tangency. That means that ∠APO = ∠BPO = 90º.
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
What do the angles in a triangle add up to?
∠OBA = 27º
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
The radius is 13.2 cm.
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.
What is 9000 m in km?
9000 m = 9 km
a2 + b2 = c2
a = ?
b = 6400 km
c = 6400 + 9
= 6409 km
The distance to point H is 340 km.
Follow the steps outlined on page 392.
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 bisector of a chord in a circle passes through the centre of the circle.
The perpendicular from the centre of a circle to a chord bisects the chord.
A line that joins the centre of a circle and the midpoint of a chord is perpendicular to the chord.
Point O is the centre of a circle, and line segment OC bisects chord AB.
∠OAC = 33º
Determine the values of xº and yº.
What type of triangle is ΔOAB?
xº = 33º, yº = 57º
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
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
CD = 8.307 X 2 = 16.6 cm
Follow the steps outlined on page 404-405.
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.
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
All inscribed angles subtended by a semicircle are right angles (90º).
Determine the values of xº and yº.
Which angles are central angles and which are inscribed angles?
xº = 55º
yº = 110º
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.
The rectangle is 13.7 cm long.
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?
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?
What is the measure of yº?
The distance from the centre of a circle to any point on its circumference.
The angle formed by joining the endpoints of an arc to the centre of the circle.
Point of tangency
The angle formed by joining the endpoints of an arc to a point on the circle.