Math 9 – Unit 8 Circle Geometry.

Math 9 – Unit 8 Circle Geometry

Unit 8 – Circle Geometry 8.0 – Introduction & definitions
8.1 – Properties of Tangents to a Circle 8.2 – Properties of Chords in a Circle 8.3 – Properties of Angles in a Circle

8.0 – Introduction & Definitions
Diameter The distance across a circle, measured through its centre Radius The distance of line segment from the centre of a circle to any point on the circle Circumfrence is…..

Chord(s) A line segment that joins two points on circle Arc A segment of the circumference of a circle Major Arc – the longer of two arcs Minor Arc – the shorter of two arcs

Tangent A line that intersects a circle at only one point Point of Tangency The point where a tangent intersects a circle

Central Angle An angle whose arms are radii of a circle Inscribed Angle An angle in a circle with its vertex and endpoints of its arms on the circle

Circle Properties that relate: A tangent to a circle and the radius of the circle A chord in a circle, its perpendicular bisector, and the centre of the circle The measures of angles in circles You will need a ruler, I will supply protractors you will want your own compass Knowing the properties of circles and the lines that intersect them helps us to use circles in designs, to calculate measurements involving circles, and to understand natural objects that are circular.

8.1 – Properties of Tangents to a Circle
Consider a bike wheel on a flat surface What angle does the spoke meet the ground? The ground is a tangent line to the wheel

Tangent-Radius Property A tangent to a circle is perpendicular to the radius at the point of tangency <APO = <BPO = 90o

Point O is the centre of a circle and AB is tangent to the circle. In triangle OAB the <AOB = 55o Determine the measure of <OBA Use Angle Sum Theory = 145 180 – 145 = 35

You will need to be able to apply Pythagorean theorem to solve problems Remember: a2 + b2 = c2 First identify the hypotenuse (or C) C = 5

Determine the length of the radius to the nearest tenth Hint – First identify properties you know – tangent, radius, angle, hypotenuse

Putting it all together: An airplane (A) is cruising at an altitude of 9000m. A cross section of Earth is a circle with a radius of about 6400 km. A passenger wonders how far she is from a point H on the horizon she sees outside the window. Calculate the distance to the nearest km. Hint – DRAW the problem

Word Problem Example: Altitude of 9000m Radius of about 6400 km How far from a point H on the horizon Calculate the distance to the nearest km. 9000 m = 9 km Use Pythagorean to calculate AH = (or 340km from the horizon)

Math Practice Page 388 – 389 Questions – 3, 4, 5, 6, 7, 8, 13, and 16c

8.2 – Properties of Chords in a Circle
Trace a circle using half a petri dish Cut it out Choose any two points on your circle NOT through the centre and draw a straight line between them Label your line segment AB. This is called a chord Fold your circle so that A coincides with B Open your circle and draw a straight line along the crease that was formed Label the point where your crease line intersects AB as point C

What observations can you make? About the angles formed? About the length of AC and CB? Are your observations true for other students? What if you repeat it with two different points?

Remember that a chord is line segment that joins two points on a circle Chord Property 1 – perpendicular from the centre of a circle to a chord bisects the chord Chord Property 2 – The perpendicular bisector of a chord in a circle passes through the centre of a circle Chord Property 3 – A line that joins the centre of a circle and the midpoint of a chord is perpendicular to the chord

Perpendicular means there is a 90o angle Bisector means it is divided into 2 equal parts What does that mean? If AC = 10cm then BC = 10cm

Put it use easy example Find the unknown angles Recognize the radius and its relationship to the isosceles triangle = 123 180 – 123 = 57

Identify the unknown angles X = 90o Y = 50o

Identify the unknown angles Remember that in an isosceles triangle the 2 base angles are equal X = 90 Y = 35 Z = 55

Put it to use Medium example Find the length of CE Where are the triangles? Diameter is 26 so radius is half that Radius 13 is hypotenuse Use pythagorean to get CD and then double it to get CE

Draw and Solve The diameter of a circle is 18cm. A chord JK is 5cm from the centre. Find the length of the chord. Step 1 – DRAW the circle Step 2 – Consider properties/relationships you know (diameter/radius, pythagorean)

Does your circle look like this? (ish) A2 + b2 = c2 25 + b2 = 81 b2 = 81-25 b2 = 56 b =

Math Practice 8.2 – Page 397 – 398 Questions 4, 5, 6, 7 and 10

Practice Problems

8.3 – Properties of Angles in a Circle
Draw a circle, cut it out Choose 2 points on the circle and label them A & B Choose a 3rd point C and connect AC and BC Fold your circle in half two times (this should make a cone), unfold it. Use the creases to identify the centre. Label the centre of the circle O and connect AO and BO Measure angles AOB and ACB What do you notice about your angles? Compare with peers.

The measure of a central angle is ALWAYS twice the measure of an inscribed angle subtended by the same arc X = 35 Y = 36

A section of the circumference of a circle is an arc The shorter arc AB is the minor arc The longer arc AB is the major arc

The angle formed by joining the endpoints of an arc to the centre of the circle is the central angle The angle formed by joining the endpoints of an arc to a point on the circle is an inscribed angle Both the inscribed and central angles in this circle are subtended by the minor arc AB

Central Angle and Inscribed Angle Property 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. True for ANY inscribed angle.

The 2 arcs formed by the endpoints of a diameter are semicircles. Central angle – 180o Inscribed angle – 90o The central angle of each arc is a straight angle, which is 180°. The inscribed angle subtended by a semicircle is one-half of 180°, or 90°.

Angles in a Semicircle Property All inscribed angles subtended by a semicircle are right angles We say: The angle inscribed in a semicircle is a right angle. We also know that if an inscribed angle is 90°, then it is subtended by a semicircle.

Inscribed Angles Property In a circle, all inscribed angles subtended by the SAME arc are congruent.

Circle geometry is all about putting the properties to work Puzzle solving! What do you know about inscribed and central angles that will help you solve for X and for Y? X – 55 Y - 110

Rectangle Puzzle 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? Join C – A then you can apply angles in a semicircle property Length – 13.7 cm long

Math Practice 8.3 – Page 410 – 411 Questions 3, 4, 5, 6, 7 and 11

Math 9 – Unit 8 Circle Geometry.

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Math 9 – Unit 8 Circle Geometry.

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