Secondary 2
Chapter 9, 10 & 11
Secondary II
Chapter 9 – Circles
Chapter 10 – Arcs and Sectors of Circles
Chapter 11 – Volume of 3-dimensional Shapes & Cavalieri’s Principle
2014/2015
Date
Section
Assignment
Concept
A: 12/10
B: 12/11
9.1, 9.2 &
9.3
- Worksheet 9.1 – 9.3
Introduction to Circles
Central Angles, Inscribed Angles, and
Intercepted Arcs
A: 12/12
B: 12/15
9.4 & 9.5
- Worksheet 9.4 – 9.5
Chords
Tangents and Secants
A: 12/16
B: 12/17
10.1 & 10.2
- Worksheet 10.1 - 10.2
Inscribed and Circumscribed Triangles and
Quadrilaterals
A: 12/18
B: 12/19
10.3
- Worksheet 10.3
Sectors and Segments of a Circle
12/22 – 1/2
A: 1/5
B: 1/6
Winter Break
11
- Worksheet 11
A: 1/7
B: 1/8
Review
A: 1/9
B: 1/12
TEST (9, 10 & 11)
Volume of cones, pyramids, cylinders, &
spheres
Cavalieri’s Principles
A: 1/13
B: 1/14
A: 1/15
B: 1/16
SAGE Testing
Late and absent work will be due on the day of the review (absences must be excused). The review assignment
must be turned in on test day. All required work must be complete to get the curve on the test.
Remember, you are still required to take the test on the scheduled day even if you miss the review, so come
prepared. If you are absent on test day, you will be required to take the test in class the day you return. You will
not receive the curve on the test if you are absent on test day unless you take the test prior to your absence.
1
Secondary 2
Chapter 9, 10 & 11
2
Secondary 2
Chapter 9, 10 & 11
Chapter 9: Circles
9.1/9.2/9.3 – Introduction to Circles, Central Angles, Inscribed Angles and Intercepted Arcs and
Measuring Angles Inside and Outside of Circles
Circle: the set of all points in a plane that are equidistant from a
given point, called the center.
Radius: the distance from the center to a point on the circle.
Chord: a segment whose endpoints are points on the circle.
Diameter: the distance across a circle, through the center. The
diameter is twice the radius.
Secant: a line that intersects a circle in two points.
Tangent: a line in the plane of a circle that intersects the circle in
exactly one point.
Point of Tangency: the point at which the tangent line intersects the circle.
Central Angle: an angle whose vertex is the center of the circle.
Inscribed angle: an angle whose vertex is on the circle.
Example 1: Four friends are riding a Ferris wheel in the positions shown.
a. Draw a central angle where Dru and Marcus are located
on the sides of the angle.
b. Draw an inscribed angle where Wesley is the vertex and
Dru and Marcus are located on the sides of the angle.
c. Compare and contrast these two angles.
3
Secondary 2
Chapter 9, 10 & 11
Arc of a Circle: any unbroken part of the circumference of a circle. An
A
arc is named using its two endpoints. The symbol used to describe
̂.
arc AB is 𝐴𝐵
D
B
Major Arc: the largest arc formed by a secant and a circle. It goes
E
more than halfway around a circle.
Minor Arc: the smallest arc formed by a secant and a circle. It goes
less than halfway around a circle.
C
Semicircle: exactly half of a circle.
Use the diagram shown to answer Questions 1 through 7.
1.
Name a diameter.
2.
Name a radius.
3.
Name a central angle.
4.
Name a tangent.
5.
Name a minor arc.
6.
Name a major arc.
7.
Name a semicircle.
Example 2: Classify each arc as a major arc, a minor arc, or a semicircle.
a.
b.
c.
4
Secondary 2
Chapter 9, 10 & 11
The degree measure of a minor arc is the same as the degree measure of its central angle.
̂ = ____________.
Example 3: In circle Z, ∠𝑋𝑍𝑌 is a central angle measuring 120°. So, 𝑚𝑋𝑌
Adjacent Arcs: two arcs of the same circle sharing a common endpoint.
Example 4: In circle A, arcs BC and CD are adjacent arcs.
̂
Give the missing arc measurements, then find 𝑚𝐵𝐶𝐷.
Arc Addition Postulate states: “The measure of an arc formed by two adjacent arcs is the sum of the
measures of the two arcs.”
Investigation:
1. Construct a circle and label its center P.
2. Use a straightedge to draw a central angle
∠𝑅𝑃𝑆.
3. Measure ∠𝑅𝑃𝑆 (in degrees) _________
4. Mark a point T and a point V, on the circle.
5. Use a straightedge to create inscribed
angles ∠𝑅𝑇𝑆 and ∠𝑅𝑉𝑆
6. Measure ∠𝑅𝑇𝑆 ___________ and ∠𝑅𝑉𝑆
____________
Compare your results. What patterns did you find in the measures of the 3 angles measured?
The Inscribed Angle Theorem states: “The measure of an inscribed angle is one half the measure of
its intercepted arc.”
5
Secondary 2
Chapter 9, 10 & 11
Example 5: In circle M, ∠𝐽𝐾𝐿 is an inscribed angle whose intercepted arc JL measures 66°.
𝑚∠𝐽𝐾𝐿 = ______________.
Parallel Line-Congruent Arcs Theorem states: “Parallel lines
intercept congruent arcs on a circle.”
̂=
Example 6: Lines PA and LR are parallel lines on circle O and the 𝑚𝑃𝐿
̂ = __________
13°. The measure of 𝐴𝑅
Example 7:
1.Determine the measure of each minor arc.
a.
2. Determine the measure of each central or inscribed angle.
b.
3. The measure of ∠GOE is 74º
a.
b.
4.
What is the measure of ∠GFE?
6
Secondary 2
Chapter 9, 10 & 11
The vertex of an angle can be located inside of a circle, outside of a circle, or on a circle. In this lesson, you
will explore these locations and prove theorems related to each situation.
The Interior Angles of a Circle Theorem states: “If an angle is formed by two intersecting chords or
secants of a circle such that the vertex of the angle is in the interior of the circle, then the measure of
the angle is half of the sum of the measures of the arcs intercepted by the angle and its vertical
angle.”
Example 8: In circle P, chords QR and ST intersect to form vertex angle TVR
1
̂ ). Find 𝑚∠𝑇𝑉𝑅.
̂ + 𝑚𝑄𝑆
and its vertical angle QVS. So, 𝑚∠𝑇𝑉𝑅 = 2 (𝑚𝑇𝑅
The Exterior Angles of a Circle Theorem states: “If an angle is formed
by two intersecting secants, two intersecting tangents, or an intersecting
tangent and secant such that the vertex of the angle is in the exterior
of the circle, then the measure of the angle is half the difference of the
measures of the arc(s) intercepted by the angle.”
Example 9: In circle C, secant FH and tangent FG intersect to form
1
̂ − 𝑚𝐽𝐺
̂ ). Find 𝑚∠𝐺𝐹𝐻.
vertex angle GFH. So, 𝑚∠𝐺𝐹𝐻 = 2 (𝑚𝐺𝐻
Investigation:
a. Create a circle,
b. Add a tangent line,
c. Add a radii that touches
the point of tangency,
d. Measure the angle formed.
e. What did you discover?
7
Secondary 2
Chapter 9, 10 & 11
The Tangent to a Circle Theorem states: “A line drawn tangent to a circle is perpendicular to
a radius of the circle drawn to the point of tangency.”
Example 10: Radius ̅̅̅̅
𝑂𝑃 is perpendicular to the tangent line s.
What is missing from the picture?
Example 11: Molly is standing at the top of Mount Everest, which has an
elevation of 29,029 feet. Her eyes are 5 feet above ground level. The
radius of Earth is approximately 3960 miles. How far can Molly see on
the horizon?
Example 12:
1. Write an expression for the measure of the given angle.
a.
b.
2.
List the intercepted arc(s) for the given angle.
3.
8
Secondary 2
Chapter 9, 10 & 11
Additional Examples
9
Secondary 2
Chapter 9, 10 & 11
9.4/9.5 – Chords, Tangents and Secants
(Standard: G.C.2)
Chords and their perpendicular bisectors lead to several interesting conclusions. In this lesson, we will
prove theorems to identify these special relationships.
Consider circle C with points B, Y, and R.
̅̅̅̅ .
a. Draw chord 𝑌𝑅
b. Construct the perpendicular bisector of chord YR.
c. Draw chord ̅̅̅̅
𝐵𝑅.
d. Construct the perpendicular bisector of chord BR.
e. What do you notice about the relationship between the
perpendicular bisectors of a chord and the center point
of the circle?
The Diameter–Chord Theorem states: “If a circle’s diameter is perpendicular to a chord, then the
diameter bisects the chord and bisects the arc determined by the chord.”
Use circle T to draw two congruent chords that are not parallel to each other and do not pass through the
center point of the circle.
a.
Construct the perpendicular bisector of each chord.
b.
Use your compass to compare the distance each chord
is from the center point of the circle.
c.
What do you notice?
The Equidistant Chord Theorem states: “If two chords of the same circle or congruent circles are
congruent, then they are equidistant from the center of the circle.”
The Equidistant Chord Converse Theorem states: “If two chords of the same circle or congruent
circles are equidistant from the center of the circle, then the chords are congruent.”
10
Secondary 2
Chapter 9, 10 & 11
Consider circle C shown.
a. Draw two congruent chords.
b. Draw four radii by connecting the endpoints of
each chord with the center point of the circle.
c. What do you observe?
The Congruent Chord–Congruent Arc Theorem states: “If two chords of the same circle or congruent
circles are congruent, then their corresponding arcs are congruent.”
The Congruent Chord–Congruent Arc Converse Theorem states: “If two arcs of the same circle or
congruent circles are congruent, then their corresponding chords are congruent.”
Segments of a chord are the segments formed on a chord when two chords of a circle intersect.
Consider circle C.
a. Draw two intersecting chords such that one chord connects two
primary colors and the second chord connects to secondary
colors.
b.
Label the point at which the two chords intersect point E.
c. Use a ruler to measure the length of each segment on the two
chords.
d. What did you discover?
The Segment–Chord Theorem states that “if two chords in a circle intersect, then the product of the
lengths of the segments of one chord is equal to the product of the lengths of the segments of the
second chord.”
Find the value of x by using the Segment-Chord Theorem.
1.
2.
11
Secondary 2
Chapter 9, 10 & 11
Previously, you proved that when a tangent line is drawn to a circle, a radius of the circle drawn to the
point of tangency is perpendicular to the tangent line. This lesson focuses on tangent lines drawn to a circle
from a point outside the circle.
Consider point P located outside of the Moon, Earth, and the Sun. Lines AF and BE are drawn tangent to the
Moon, Earth, and the Sun as shown.
A tangent segment is a line segment formed by connecting a point outside of the circle to a point of
tangency.
a. Identify the two tangent segments drawn from point P associated with the Sun. Then, use a
compass to compare the length of the two segments
b. Identify the tangent segments drawn from point P associated with the Moon. Then, use a compass
to compare the length of the two line segments.
c. Compare your findings from a and b.
The Tangent Segment Theorem states: “If two tangent segments are drawn from the same point on
the exterior of a circle, then the tangent segments are congruent.”
12
Secondary 2
Chapter 9, 10 & 11
A secant segment is the line segment formed when two secants intersect outside a circle. A secant
segment begins at the point at which the two secants intersect, continues into the circle, and ends at the
point at which the secant exits the circle.
An external secant segment is the portion of
each secant segment that lies on the outside of
the circle. It begins at the point at which the
two secants intersect and ends at the point
where the secant enters the circle.
The Secant Segment Theorem states: “If two secants intersect in the exterior of a circle, then the
product of the lengths of the secant segment and its external secant segment is equal to the product
of the lengths of the second secant segment and its external secant segment.”
Example: In circle C, above, secant segments ̅̅̅̅
𝐷𝐴 and ̅̅̅̅
𝐸𝐵 intersect at point P outside the circle.
̅̅̅̅
̅̅̅̅ ∙ 𝐴𝑃 = 𝐸𝑃
̅̅̅̅ ∙ ̅̅̅̅
So, 𝐷𝑃
𝐵𝑃
The Secant Tangent Theorem states: “If a tangent and a secant intersect in the exterior of a circle,
then the product of the lengths of the secant segment and its external secant segment is equal to the
square of the length of the tangent segment.”
Example 1: In circle C, tangent ̅̅̅̅
𝐴𝑃 and secant ̅̅̅̅
𝐸𝐵 intersect at
point P outside the circle.
So, ̅̅̅̅
𝐵𝑃 ∙ ̅̅̅̅
𝐸𝑃 = ̅̅̅̅
𝐴𝑃2
13
Secondary 2
Chapter 9, 10 & 11
Example 2:
1. If the length of AB is 24 cm, what is the length of CD?
2.
3. Find PQ
a.
4.
b.
Solve for x.
a.
b.
14
Secondary 2
Chapter 9, 10 & 11
Additional Examples
15
Secondary 2
Chapter 9, 10 & 11
Chapter 11: Volume of 3-dimensional shapes & Cavalieri’s Principle
Example 1: Volume of a Cylinder. Formula: _____________________
Find the volume of the following cylinder.
Example 2: Volume of a Cone. Formula: ______________________
Find the volume of the following cone.
a.
b.
Example 3: Volume of a Pyramid. Formula: ______________________
a. Find the volume of the square pyramid.
b. Find the volume of the triangular pyramid.
16
Secondary 2
Chapter 9, 10 & 11
Example 4: Volume of a sphere. Formula: _______________________
Find the volume of each sphere.
a.
b.
Applying Cavalieri’s Principles
Cavalieri’s principle for 2-dimensional figures states that if the lengths of one-dimensional slices- just a line
segment- of two figures are the same, then the figures have the same area.
Cavalieri’s principle for 3-dimensional figures states that if, in two solids of equal altitude, the sections
made by planes parallel to and at the same distance from their respective bases are always equal, then the
volumes of the two solids are equal.
Example 5: Estimate the area or volume of each figure.
a.
b.
17
Secondary 2
Chapter 9, 10 & 11
Additional Examples
18
Secondary 2
Chapter 9, 10 & 11
Chapter 10: Arcs and Sectors of Circles
10.1/10.2 – Inscribed and Circumscribed Triangles and Quadrilaterals and Arc Length
(Standard: G.C.3)
An inscribed polygon is a polygon drawn inside a circle such that each vertex of the polygon touches the
circle.
Consider ∆𝐴𝐵𝐶 that is inscribed in circle P.
1. What do you know about ̅̅̅̅
𝐴𝐶 ?
̂ ? Explain your reasoning.
2. What do you know about 𝑚𝐴𝐷𝐶
3. What does this tell you about 𝑚∠𝐴𝐵𝐶? Explain your reasoning.
4. What kind of triangle is ∆𝐴𝐵𝐶? How do you know?
Draw a triangle inscribed in the circle through the three points.
Then determine if the triangle is a right triangle.
a.
b.
c. In ∆𝐴𝐵𝐶, if 𝑚∠𝐶 = 62°, can
you determine 𝑚∠𝐴 ?
The Inscribed Right Triangle–Diameter Theorem states: “If a triangle is inscribed in a circle such that
one side of the triangle is a diameter of the circle, then the triangle is a right triangle.”
The Inscribed Right Triangle–Diameter Converse Theorem
states: “If a right triangle is inscribed in a circle, then the
hypotenuse is a diameter of the circle.”
19
Secondary 2
Chapter 9, 10 & 11
Example 1: Consider the relationship between opposite angles of an inscribed quadrilateral.
a. Use your compass to draw a circle.
b. Use your straightedge to draw an inscribed
quadrilateral that is not a parallelogram in
your circle. Label the vertices of your
quadrilateral.
c. Use your protractor to determine the
measures of the angles of the quadrilateral.
What is the relationship between the measures
of each pair of opposite angles?
The Inscribed Quadrilateral–Opposite Angles Theorem states: “If a quadrilateral is inscribed in a
circle, then the opposite angles are supplementary.”
Example 2:
a. In quadrilateral ABCD, 𝑚∠𝐵 = 112°.
Determine 𝑚∠𝐷.
b. In quadrilateral ABCD, 𝑚∠𝐴 = 72°°.
Determine 𝑚∠𝐶.
A circumscribed polygon is a polygon drawn
outside a circle such that each side of the
polygon is tangent to a circle.
A circle can be inscribed in a quadrilateral if and only if the angle bisectors of the four angles of the
quadrilateral are concurrent.
20
Secondary 2
Chapter 9, 10 & 11
So far you have described the measures of angles and arcs using degrees. A radian is another unit that can
be used to measure angles and arcs.
Example 3: Plot the following angles on the coordinate system.
a.
3
4
b.
y
5𝜋
8
6𝜋
5
y
x
e.
c.
5
3
3
2
y
x
g. 2.861
y
x
9𝜋
4
y
x
f.
y
d.
h.
y
x
x
6.0
y
x
x
21
Secondary 2
Chapter 9, 10 & 11
Additional Examples
22
Secondary 2
Chapter 9, 10 & 11
10.3 – Arc Length, Sectors and Segments of a Circle
(Standard: G.C.5)
Example 1: Convert the following degrees to radians.
a. 45°
b. −660°
__________
__________
c. −270°
__________
d. 120°
___________
Example 2: Calculate the arc length of each circle. Express your answer in terms of 𝜋.
23
Secondary 2
Example 3: If 𝜃 =
Chapter 9, 10 & 11
𝜋
2
and 𝑟 = 4, solve for the length of the intercepted arc.
Example 4: If 𝑟 = 2 and the intercepted arc length is 5, what is the measure of the central angle?
The Sector of a Circle is a region of the circle
bounded by two radii and the included arc.
To determine the area of a sector, use the following formula:
Area of a sector =
𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑎𝑛𝑔𝑙𝑒
360°
∙ 𝜋𝑟 2 , where r is the length of the radius of the circle.
Example 5: Find the area of the sector.
Example 6: Calculate the area of each sector. Write your answer in terms of 𝜋.
a. If the radius of the circle is 9 centimeters,
what is the area of sector AOB?
b. If the radius of the circle is 15 feet,
what is the area of sector EOF?
24
Secondary 2
Chapter 9, 10 & 11
A segment of a circle is a region of the circle bounded by a chord and the included arc.
Area of a segment = Area of a Sector – Area of a Triangle
Example 7:
Calculate the area of each segment. Round your answer to the nearest tenth, if necessary.
1. If the radius of the circle is 6 centimeters,
what is the area of the shaded segment?
2. If the radius of the circle is 25 meters, and
∠𝐼𝑂𝐽 = 60° what is the area of the shaded
segment?
25
Secondary 2
Chapter 9, 10 & 11
Additional Examples
26