Download Mathematics 9 Unit #3: Shape and Space Sub

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Mathematics 9
Unit #3: Shape and Space
Sub-Unit #3: Circle Geometry
Lesson
1
Topic
Triangles
2
Tangents
3
Chords
4
Angles
5
Situational
Problems
Quiz #5
“I Can”
 Recall that the interior angles of a triangle have a sum of 180°
 Solve for an interior angle of a triangle, given the other two
 Explain properties of side length of isosceles triangles
 Explain properties of angle sizes of isosceles triangles
 Explain properties of side length of equilateral triangles
 Explain properties of angle sizes of isosceles triangles
 Use Pythagorean Theorem to solve for the hypotenuse of a right
triangle, given both legs
 Use Pythagorean Theorem to solve for a leg of a right triangle,
given the hypotenuse and one leg
 Describe the relationship between tangents and the radius of a
circle.
 Construct tangents by applying the relationship between the
radius of a circle and the tangents
 Use triangle properties to solve for an unknown quantity
 Describe the relationship between a chord, its perpendicular
bisector and the center of a circle of a circle
 Use triangle properties to solve for an unknown quantity
 Understand the relationship between inscribed angles and
central angles
 Understand the relationship between inscribed angles
subtended by the same arc
 Find the diameter of a circle using an inscribed angle of 90⁰
 Use triangle properties to solve for an unknown quantity
 Solve situational problems using properties of circles and
triangles
Demonstrate your understanding of circle properties
Lesson #1 – Triangles
In any triangle the sum of the interior angles is 180 . This means if you have the measure of
two angles, implicitly you have the measure of the third.
Ex) Solve for the unknown angle:
Another special property for right angle triangles is the Pythagorean Theorem that states that
the sum of the squares of the legs is equal to the square of the hypotenuse. A right angle
triangle is a special triangle that has one 90⁰ angle. The two sides straddling the 90⁰ angle are
called “legs” while the side opposite the angle is called the hypotenuse.
Ex2) Label the legs and the hypotenuse of this triangle:
The Pythagorean States: the Sum of the squares of the legs equals the square of the
hypotenuse:
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For the following triangles, state the Pythagorean Theorem:
To solve for an unknown side, determine the hypotenuse and the legs, plug in the two given
values and solve for the third, using your rules for solving an equation:
Ex3) Solve for the unknown side in each of the following:
a)
b)
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c)
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Assignment:
Lesson #2 – Tangents
Using the circle to the right:
1. Draw any chord on the circle using a ruler. DO
NOT PICK UP THE RULER!
2. Move your ruler away from the line slowly.
Look at the two points that the ruler intersects
the circle. When you hit a point where the
ruler is only touching the line once, draw
another line.
The tangent is a line that intersects the circle at only one point, no matter how much it is extended.
The point of tangency is that point at which a tangent intersects a circle.
Use the circle below:
1. Sketch a tangent anywhere on this circle.
2. Draw a radius from the center, O, of the circle to
the tangent you have drawn
3. Measure the angle between the radius and the
tangent. Compare with your neighbor.

What is the angle between the radius and the
tangent?

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tangent for your nearest neightbor?
What is the angle between the radius and the
CIRCLE PROPERTY #1
A tangent to a circle is perpendicular to the radius at the same point of tangency.
Use this property to solve the following; assume O to be the center of every circle:
Ex1) Solve for
Ex2) Solve for x.
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Ex3) Solve for the indicated length; round any necessary answers to one decimal place
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Assignment: Page 388-390 # 3-7, 12, 14
Lesson #3 – Chords
A chord is any line that intersects a circle twice. Sketch chords on the following circles.
Draw a chord on the piece of paper on the next page and remove it from the booklet. Fold the paper so
that the two edges of the chord coincide with each other. In the space below, write your observations:

What is special about the fold line? What is this called?

Measure the angle between your fold line and your chord. What about your neighbors?

Measure the length of the line from the fold line to the outside edge of the circle on both sides of
the fold line (2 measurements). What do you notice about these lengths? What does your neighbor
notice?
The line of symmetry is called a _____________________ _________________. A line bisects another
line when it goes through the midpoint of the line (____________ ______ ______ ____________).
Therefore a ____________________ _________________ is a line that goes through the midpoint of a
chord and is perpendicular to the chord. This leads us to the circle properties
CIRCLE PROPERTY #2
The perpendicular from the center of a circle to a chord bisects the chord; that is, the perpendicular
divides the chord into two equal parts. Using the diagram to the right as a
reference:
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When
, then
**If you know a line from the center of the circle is perpendicular to a chord,
then it also bisects the chord.**
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CIRCLE PROPERTY #3
The perpendicular bisector of a chord in a circle passes through the center of
the circle.
Ie: When
center of the circle.
and
, then SR passes through O, the
**If a line is perpendicular to a chord and bisects the chord, then the line goes
through the center of the circle**
CIRCLE PROPERTY #4
A line that joins the center of a circle and the midpoint of a chord is
perpendicular to the chord.
Ie: When O is the center of a circle and
, then
**If a line goes through the center of the circle and the midpoint of a
chord, then it is perpendicular to the chord.
SUMMARY:
These three circle properties are like “buy two get one free”. If you are given two bits of information
you are expected to deduce the third, as they are a package deal that will always go together. You will
be looking for ways to apply them as well as your knowledge of the relationship between diameter and
radius and the Pythagorean Theorem.
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Ex1) Determine the values of
and
using circle properties.
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Ex 2) From the diagram below solve for the length of the chord CD.
Assignment: Page 397-399 # 3-7
Lesson #4 – Angles
A section of the circumference of a circle is called an
arc. There are two types of arcs, major and minor.
The _________ arc is the minor arc.
The _________ arc is the major arc.
The angle formed by joining the endpoints of an arc to
the center of the circle is called a CENTRAL ANGLE. In
the diagram below,
is a central angle.
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The angle formed by
joining the enpoinsts
of an arc to a point on
the circle is an
INSCRIBED ANGLE. In
the diagram below, ,
is an inscribed angle. Below are 2 pictures of inscribed angles and
central angles. Use your protractor to measure the angles on these and
the diagram to the left. Look for a pattern between inscribed angles an
central angles.
The measure of the central angle tends to be _________________ the measure of the inscribed angle;
or the measure of the inscribed angle tends to be _________________ the measure of the central angle.
CIRCLE PROPERTY #5
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.
(
) or
(
)
**Central angles are twice the size of the inscribed angle based on the
same arc.
CIRCLE PROPERTY #6
In a circle, all inscribed angles subtended by the same arc are all
congruent.
**All inscribed angles are equal if based off the same arc
CIRCLE PROPERTY #7
All inscribed angles subtended by a semicircle are right angles.
Since
, then it follows that
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**If an inscribed angle is subtended by the diameter, then the
angle is a right angle or 90⁰, since the central angle is 180⁰.
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Ex1) Triangle ABC is inscribed in a circle with the center O.
Determine the values of , , and .
and
.
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Ex2) Rectangle ABCD has its vertices on a
circle with a radius of 8.5 cm, as pictured to
the right. The width of the rectangle is 10.0
cm. What is its length? Give the answer to
the nearest tenth of a cm.
Assignment: Page 410-411 # 3-6; 11
Lesson #5 – Situational Problems
There are many situational problems that can be solved using the circle properties explained throughout
this unit. The hardest part of solving these problems is determining which property or properties you
must use.
The most important key to solving situational problems is knowing the properties and the terms
involved in the properties. The net key to solving situation problems is drawing a diagram and
understanding which values you are looking for. Luckily, in grade 9 you will be given the diagrams!
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Ex1) A small aircraft, A, is cruising at an altitude
of 1.5 km. The radius of the Earth is
approximately 6400 km. How far is the plane
from the horizon at point B? Calculate his
distance to the nearest km.
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Ex 2) A radar station, R, tracks all ships
in a circle with a radius of 50.0 km. A
ship enters this radar zone ant the
station tracks it from 62.5 km, until the
ship passes out of range. What is the
closest distance the ship comes to the
radar station?
Assignment: Page 418-419 # 2, 3, 8, 10,