Download Unit 6 - Katey Parham

Warm Up
Apr. 24th
1. The perimeter of an equilateral triangle is 36 inches. Find the
length of an altitude. Give the exact answer (in simplified
radical form).
2. a = ____
3. b = ____
4. Find the exact value of x.
x
15
3 10
Homework Check/Questions??
Thurs. Apr. 24th
Tangents & Chords
A circle is the set of all points in a plane
at a given distance (radius) from a given
point (center) in the plane.
Congruent circles –
Concentric circles -
Chord –
Secant -
Tangent line(segment) -
a. Draw a tangent line and a radius to the
point of tangency.
b. Describe the relationship between the
tangent line and the radius of the circle
drawn to the point of tangency.
ED is tangent to the circle. x = ____
AB is tangent to  C.
What is the radius of the circle?
mOJT = 30º, JO = 20, then JT = ______
JK = 9, KO = 8, then JT = ______
Theorem: The tangent segments to a circle
from a point outside the circle are congruent.
How do you know?
Example: In the diagram, RT = 12 cm, RH = 5 cm,
and MT = 21 cm. Determine the length of MR .
In the circle:
• Draw a diameter.
• Draw a chord that is perpendicular to the diameter.
What appears to be true of a chord perpendicular to a
diameter?
Example: A chord of a circle is 12 in. long, and its
midpoint is 8 in. from the center of the circle.
Calculate the length of the radius of the circle.
Theorem: In a circle (or congruent circles), two
congruent chords are equidistant from the
center of the circle.
RS = ______
1)
2)
3)
4)
Start Thurs.
Apr. 24th & Finish
Fri. Apr. 25th
Angles and Arcs
Vocab & Chart
An arc is an unbroken part of a circle.
Three types of arcs…
•Minor Arc -less than half of the circle
•Semi-circle- exactly half of the circle
•Major Arc -more than half of the circle.
A central angle of a circle is an angle with
its vertex at the center of the circle.
Type of angle
Central Angle
Where is the
vertex?
Example
Description of angle
measurement
Formula for measure
using example
An inscribed angle is an angle whose vertex is
on the circle and whose sides contain chords
of the circle.
Type of angle
Inscribed
Angle
Formed by
tangent and
chord
Where is the
vertex?
Example
Description of angle
measurement
Formula for measure
using example
“inside” angle…
Type of angle
“Inside” angle
Where is the
vertex?
Example
Description of angle
measurement
Formula for measure
using example
“outside” angles
Examples
Find the measure of each arc.
Angle formed by a chord (or secant) and tangent…
You Try!
Any angle inscribed in a semi-circle is a _____ angle.
If a quadrilateral is inscribed in a circle, then the
opposite angles are supplementary.
…and the special note
Quiz Day!
Apr. 28th
1) Clear your desk except something to
write with and a calculator.
2) Please leave answers in simplified radical
form, no rounded decimals!
3) When finished, please turn it in on the
cart and check your homework answers.
Homework Check & Questions?
Mon. Apr. 28th
Lance owns The Flyright Company. His company specializes in
making parachutes and skydiving equipment. After returning
from a tour of Timberlake Gardens, he was inspired to create
a garden in the circular drive in front of his office building.
Lance decided to hire a landscape architect to design his
garden. The architect told Lance that he would need to
determine the area and circumference of the garden.
1. What formula can be used to determine the circumference of the
circle? What information does the circumference provide about the
garden?
2. What formula can be used to determine the area of the circle?
What information does the area provide about the garden?
The layout of The Flyright Company’s building
and parking lot are shown.
3. What is the radius,
circumference and area of the
largest circle that will fit in the
grassy area? Justify your
answer. Give your answers in π
form then round each to 1
decimal place.
4.
Lance decides he would like the garden to
have a circumference of 40π feet. What
would be the area of this garden?
5. If the area of the garden needs to be 225 π feet, what
would be the circumference?
6.
Lance is also considering including a sidewalk
around the outside of the garden, as shown
below. Determine the area of the sidewalk.
Lance would like his garden to resemble a colorful
parachute with different flowers in alternating areas of
the garden. His sketch for the landscaper is shown below.
The circle garden is divided into 8 equal parts (called sectors).
a)
What portion of the total area of the circle is each part?
Recall that Lance would like the diameter of his circle garden to be 20 feet.
b) Determine the area of each sector of the garden.
c) Determine the arc length of each sector of Lance’s garden.
Mrs. Little boards a Ferris wheel with a diameter
of 100 feet. Calculate the approximate distance
that Mrs. Little traveled in her 6 revolutions. Round
to 3 decimal places.
Use circle B below to answer the following:
a) Area:
b) Area of Shaded Sector:
c) Circumference:
d) Length of AC:
Examples
The length of an arc in a circle with radius 8 inches is
3.2π inches. Determine the measure of the arc.
The measure of the central angle of a sector is 60° and
the area of the sector is 6π inches. Calculate the radius
of the circle.
Warm Up
Apr. 29th
1) Find the midpoint and distance of (-3, 6)
and (7, 11)
2) Expand and simplify: (x – 3)2
3) Factor: x2 + 8x + 16
4) Solve by completing the square:
x2 – 4x + 10 = 31
Homework Check & Questions?
Tues. Apr. 29th
Revisit the definition….
A circle is the set of all points in a plane
that are equidistant from a given point.
Write the equation of the circle described …
• Center (0, 3) and radius = 5
• Center (-4, 5) and radius = 7
Write the equation of the circle described …
• Center (2, -1) and contains the point (0, 4)
• Endpoints of the diameter (-1, 3) and (5, 1)
More Circles
Write the equation of the circle described …
• Contains the three points (2, 4), (6, 2), and (-1, -5)
Identify the center and radius of the circle
• (x – 3)2 + (y + 1)2 = 49
• x2 + (y – 5)2 = 10
What would this equation look like if we
expanded the binomials….
• (x – 3)2 + (y + 1)2 = 49
• x2 + (y – 5)2 = 10
Rewrite the equation in information form and
identify the center and radius of the circle
• x2 + y2 – 6x + 2y = 6
• x2 + y2 + x + 4y = 0
Determine the area and circumference of
the circle whose equation is:
• (x – 3)2 + y2 = 12
• x2 + y2 - 10x + 8y = 8
Given circle O with equation x2 + y2 – 8y + 7 = 0
• Rewrite the equation in information form.
• Determine the center and radius of the circle.
• Determine whether each point below is in the
interior, exterior or on the circle.
– (5,-2)
– (0,7)
– (1, 3)
Warm Up
Apr. 30th
1.
Given points (-5, 2) and (-1, 6) are the
endpoints of a diameter of a circle. Write the
equation of the circle.
2. Determine the center and radius of the circle:
x2 + y2 – 6x – 10y = 2
3. QR = ____
4. QP = ____
5. TP = ____
mTPS  150 ;RP  8 3
Wed. Apr. 30th
Quadratic Functions as Conics
Homework Check & Questions?
y = ¼x2
What is the vertex?
Look at the point F(0, 1)
and the line y = -1. What
do you notice about the
distance from F to the
vertex and the distance
from the line y = -1 to the
vertex?
A parabola is the set of all points that are
equidistant from a fixed point, called the
focus, and a fixed line called the directrix. A
focus and directrix uniquely determine a
parabola.
In general, if the distance from the vertex to the
focus (or directrix) is c and the vertex is (h, k), the
equation in information form is
1
y
(x  h)2  k
4c
__________________________________
A parabola with focus F(2, 4) and directrix y = -2.
What are the
coordinates of the
vertex of the parabola?
What is the equation for the
parabola in information form?
Identify the coordinates of the
vertex,
coordinates of the focus and
equation of the directrix.
1
y   (x  3)2
16
Identify the coordinates of the
vertex,
coordinates of the focus and
equation of the directrix.
y  2(x  1)2  3
Examples: Write the equation of the parabola.
Write the equation of the parabola with vertex
(2, -3) and directrix y = -5
Warm Up
May 1st
1. Rewrite the equation in information form,
identify the center and radius:
x2 + y2 + 8x – 6y = 1
2. Graph the circle: (x – 1) 2 + y2 = 9
3. Determine the focus and directrix
y = -3(x + 1)2 - 4
Review Day