Reordering Ranganathan: Shifting behaviors, shifting priorities.
Circles Name - Amazon S3€¦ · Equations of Circles Notes Consider the graph x 2+y 2=16. Give...
Transcript of Circles Name - Amazon S3€¦ · Equations of Circles Notes Consider the graph x 2+y 2=16. Give...
Circles Name:
Notes Date:
Gustave is trying to determine where to launch his satellite so that
it will travel the longest possible distance before it has to re-enter
earth’s atmosphere. He has four potential orbits which each have a
different amount of predicted risk of the satellite surviving passing
through Earth’s atmosphere. Gustave must make a decision.
Write the definition of the term and include an image or example that represents it.
Term Definition Example
Circle
Radius
Diameter
Chord
Tangent
Circumference
Term Definition Example
Central Angle
Sector
Arc,
Minor Arc,
Major Arc
Inscribed
Angle
There are a lot of terms related to circles. Which relate to distances and which relate to angles?
Terms that refer to distances Terms that refer to angles
The Circle as a Shape The Conceptualizer!
The circle is a basic shape that you learned
about it in elementary school. Now, you are
going to get to know it on a whole different
level.
#couldntWeJustStayFriends?
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Characteristics of Circles The Conceptualizer!
All circles are similar. They are all the
“same” shape, as all regular polygons are.
The only difference is in their size.
As the distance across the circle gets larger,
so does the distance around the circle.
That is, as the diameter gets larger, the
circumference gets larger. #wheelgoesround
History of : Circumferenceπ The Conceptualizer!
The ratio of the circumference of the circle C to its
diameter d is , an irrational number π
approximated by 3.14. There is evidence has π
been discovered for 4,000 years by the Babylonians
and Egyptians.
The number is defined as . Then the π π = dC
circumference C is given by . And the dC = π
diameter is twice the radius r, so .πrC = 2
History of : Areaπ The Conceptualizer!
Imagine the circle chopped into many narrow
sectors (“pizza slices”). Arrange the slices
alternating direction. If there is an infinite number
of slices, the slices form a rectangle. Its height is
the radius r; its base is half the circumference,
.(2πr) r21 = π
Thus, the area is . This is the area h πr)(r) rb = ( = π 2
of the original circle AND the formula we use today.
If you know the radius of the circle, you can find its
circumference or area.
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Circumference Notes
Find the circumference of a circle
with a diameter of 8 cm.
Area Notes
Find the area of a circle
with a diameter of 6 cm.
Central Angles The Conceptualizer!
There are in a circle.60°3
If you take the angle between two radii. The
vertex of the angle is at the center of the
circle. It is a central angle.
The angle cuts off an arc of the same
measure.
A central angle of cuts off an arc of .0°8 0°8
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Inscribed Angles The Conceptualizer!
An inscribed angle has its vertex on the
circle. Its measure is half the central angle
that cuts off the same arc.
All inscribed angles that cut off the same
arc are congruent.
Inscribed Angles and Right Angles The Conceptualizer!
An inscribed angle that cuts off (or
“intercepts”) a semi-circle, of , has a 80°1
measure of half that -- so it is a right angle.
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Inscribed Angles Notes
If the measure of angle BOC is , find the86°
measures of arc BC and angle BDC .
Arc Length Description
The central angle intercepts an arc of the
same angle measure -- but what of the
actual distance along the circle from A to
B? That is the arc length.
It is a portion of the circumference, in
proportion to central angle and can be
solved with this proportion:
360°measure angle AOB = 2πr
measure arc length AB
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Arc Length Notes
Find the arc length AB.
Area of a Sector Description
The area between the two radii is called a
sector .
The pie slice AOB is a portion of the area of
the entire circle and can be solved with this
proportion:
360°measure angle AOB = πr2
area sector AOB
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Area of Sector Notes
Find the area of sector ACB.
Tangents of a Circle The Conceptualizer!
If a line touches a circle at only one point,
it is called a tangent.
A few rules--
(1) If a line is tangent to a circle, then it
is perpendicular to the radius drawn
to the point of tangency.
(2) The converse is true too. This
relationship often creates right
triangle Pythagorean Theorem
Questions.
(3) If two segments from the same point
are tangent to a circle, then they are
congruent.
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Applications of Tangent Notes
Segment CD is a tangent to Circle A.
What is the radius of Circle A?
A different definition of a circle.
We’ve always thought of a circle as a shape, but there is another
way to think about it --
Every point on a circle is the same distance from the center.
Then the circle is the locus of all points that are a constant
distance r from the center.
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Equation of a Circle: Origin Story The Conceptualizer!
All points on a circle are the same distance
from the center.
In this diagram, you can see 8 instances of
the 3:4:5 Pythagorean Triple right triangle.
The radius is the hypotenuse of each
triangle; it has length 5.
All the other points on the circle are 5 units
away from , too, but their coordinates 0, )( 0
are not integers.
Equation of a Circle The Conceptualizer!
Since these points on the circle can be
understood as right triangles, we can apply
the Pythagorean Theorem to understand
their coordinates and the radius.
a2 + b2 = c2
32 + 42 = 52
But it’s not just points such as or 3, )( 4
for which this equation “works”. It− ,− )( 3 4
is true for all other points, too.
That means, if is on this circle, thenx, )( y
x2 + y2 = 52
Now we have an equation for any circle:
x2 + y2 = r2
It is based on the Pythagorean Theorem and
Distance Formula.
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Equations of Circles Notes
Consider the graph . Give details 6x2 + y2 = 1
about its size and location.
Shifting the Circle The Conceptualizer!
What if you shift the center of the circle to
?2, )( 3
Now every point on the circle is 5 x, )( y
points away from . By Distance 2, )( 3
Formula:
5 = √(x )− 2 2 + (y )− 3 2
(x )− 2 2 + (y )− 3 2 = 52
The circle is .5(x )− 2 2 + (y )− 3 2 = 2
Writing Equations Notes
What is the equation for the circle with
radius 2 and center ?− , )( 4 6
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Sketch an example or an equation of each of the following.
Circumference Area of a Circle Arc Length Tangent
Jose creates a pie chart of his expenses. Of the $1,800 he spends each month, rent accounts for
$1,000. What is the central angle of the sector representing rent?
A Ferris wheel was built on Navy Pier in Chicago. Its diameter is 140 feet. How many feet does the
Ferris wheel rotate if one ride was three complete turns?
A clock has a radius of 9 cm. The sector between the hands at 4 o’clock is painted blue. What area
is covered by blue paint?
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