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Circles

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by

Tom Harrington

on 9 October 2014

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Transcript of Circles

Circles
G.11 G.12
A
circle
is the set of all of the points that are equidistant from a given point in a plane
The
radius
is the distance from the center to a point on the circle
A
chord
is a segment joining two points on the circle.
A
diameter
is a chord that passes through the circle’s center.
A
secant
is a line that contains a chord. It intersects the circle at two points
A
tangent
is a line that intersects a circle at only one point.
An
arc
is an unbroken part of a curve of a circle.
The
arc measure
is the degree measure of its central angle.
A
central angle
is an angle with its vertex at the circle’s center.
A
major arc
measures greater than 180º but less than 360º
A
minor arc
measures greater than 0º but less than 180º
Semicircles
are the two arcs of a circle that are cut off by a diameter. A semicircle measures 180
Vocabulary
The
intercepted arc
is the part of the circle that lies between the two lines that intersect the circle.
An
inscribed angle
is an angle whose vertex is on the circle and whose sides are chords of the circle.





The measure of an inscribed angle is equal to one-half the measure of its intercepted arc.
The length of an arc (
arc length
) is a linear measure




and is part of the
circumference
(perimeter of a circle).
A
sector
of a circle is that part of the circle




bounded by two radii and an arc
Central and Inscribed Angles
A
central angle
is equal to the
measure of the intersected arc
An
inscribed angle
is equal to half the measure of the intersected arc or half the measure of the
central angle with the same endpoints
Try It!
Find the measure of angle
PQR
Try It!
Find the measure of angle
PQR

120 = 2(PQR)
60 = PQR
Now go to the class
page and take the
central and inscribed
angles mini quiz
Area and Circumference
Formulas
We can use proportions
to find the arc length and
sector area of a given part
of a circle
Arc Measure / Angle

360
Sector Area / Arc Length

Circle Area / Circumference
=
Try It!
Find the arc length and area of the yellow sector
10
30
o
Try It!
Find the arc length and area of the yellow sector
10
30
o
Circumference = 2(10) = 20
Area = 10 = 100
2
Arc Length
30 x
= =
360 20
600 = 360x
x =
5
3
Arc Length
300 = 360y
30 y
= =
360 100
y =
5
6
Now go to the
school fusion page
and complete the
mini quiz on
area and circumference
Angle Relationships
When two chords intersect inside
a circle, the product of the lengths
of the segments of one chord equals
the product of the lengths of the
segments of the other chord.
The measure of an angle formed
by two chords that intersect inside
a circle is equal to half the sum of
the measures of the intercepted arcs.
Two Chords
Interior Angles
Try It!
If BP = 8, PC =6 , AP = 4,
PD = 2x - 8, find the value
of x.
Try It!
If BP = 8, PC =6 , AP = 4,
PD = 2x - 8, find the value
of x.
(BP)(PC) = (AP)(PD)

(8)(6) = (4)(2x - 8)
48 = 4(2x - 8)
12 = 2x - 8
20 = 2x
10 = x
Try It!
Find the
value of x
Try It!
Find the
value of x
x = 75 + 65
2
x = 140
2
x = 70
When segments intersect inside a circle, they also form special linear relationships
Two Secants
When two secant segments are drawn to a circle from an exterior point, the product of the lengths of one secant segment and its exterior segment is equal to the product of the lengths of the other secant segment and its exterior segment.
(PD)(PA) = (PB)(PC)
Concentric Circles
Two circles that share the same center
are called
concentric circles
We can use the properties of circles
to solve problems about concentric
circles
Ex. Find the area between the two circles
7
3
Area between = Area of Larger Circle - Area of Smaller Circle
= (7) - (3) = 49 - 9 = 40
2
2
Try It!
Find the area between the two circles
4
5
Try It!
Find the area between the two circles
4
5
(5) - (2) = 25 - 4 = 21
2
2
Linear Relationships
Secant and Tangent
When lines intersect in a circle,
they form special angle relationships
Exterior Angles
An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle.
The measure of the angle formed is equal to ½ the difference of the intercepted arcs.

Two Secants
Two Tangents
Secant and Tangent
Try It!
Find the value of x
Try It!
Find the value of x
Now go take the angle
relationship mini quiz
Try It!
Find the value of x
Try It!
Find the value of x
(AP)(CP) = (DP)(BP)
(2x+10)(10) = (2x+3+9)(9)
20x + 100 = 18x + 108
2x = 8
x = 4
The square of the length of the tangent equals the product of the length of the secant and its exterior segment.

(UV) = (UY)(UX)
2
Try It!
Find the
value of x
Try It!
Find the
value of x
10 = (3x + 5)(5)
2
100 = 15x + 25
75 = 15x
5 = x
Now go to the class site and take the linear relationships mini quiz
Now go to the class
site and take the
concentric circles mini
quiz
Equations of Circles
When a circle is drawn on a coordinate plane, it is defined by the following equation:
(x - h) + (y - k) = r

Where (h,k) is the center of the circle and
r is the radius of the circle
2
2
2
ex. The circle has a
center at (3,4) and a
radius of 5
Try It!
Write the equation of a circle that has
a center at (-2,4) and point (0,4) is on the
circle.
Try It!
Write the equation of a circle that has
a center at (-2,4) and point (0,4) is on the
circle.
(x - h) + (y - k) = r
(0 - (-2)) + (4 - 4) = r
4 + 0 = r
16 + 0 = r
16 = r
4 = r
Answer: (x + 2) + (y - 4) = 16
2
2
2
2
2
2
2
2
2
2
2
2
2
Now take the equations of
circles mini quiz
Chords and Arcs
Congruent chords create special relationships with the arcs they intersect
Theorem # 1
In a circle, if two chords are congruent then their corresponding minor arcs are congruent.

Try It!
Find the measure
of arc AB
160
40
o
o
Try It!
Find the measure
of arc AB
160
40
o
o
AB = DC
160 + 40 + AB+ DC = 360
200 + 2(AB) = 360
2AB = 160
AB = 80
Theorem # 2
In a circle, two chords are congruent if and only if they are equidistant from the.

Try It!
Given that AD = AE
CB = 2x +1
GF = 4x + 19
Find the length of GF
Try It!
Given that AD = AE
CB = 2x +1
GF = 4x - 19
Find the length of GF
2x + 1 = 4x - 19
2x + 18 = 4x
18 = 2x
9 = x
GF = 4x - 19
GF = 4(9) - 19
GF = 36 - 19
GF = 17
Theorem # 3
In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc.

Try It!
If AP = 12 and arc ACB = 140,
Find the length of AB and
the measure of arc CB
Try It!
If AP = 12 and arc ACB = 140,
Find the length of AB and
the measure of arc CB
OC bisects AB and arc ACB, so
AB = 2(AP) and ACB = 2(CB)
AB = 2(AP)
AB = 2 (12)
AB = 24
ACB = 2(CB)
140 = 2CB
70 = CB
Now go to the class site and take the
chords and arcs mini quiz
Tangents
Theorem # 1
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

Try It!
If OP = 18 and PT = 24,
what is the length of OT?
Try It!
If OP = 18 and PT = 24,
what is the length of OT?
OT = OP + PT
OT = 18 + 24
OT = 324 + 576
OT = 900
OT = 30
2
2
2
2
2
2
2
2
Theorem #2
Try It!
If PR = 3x + 5
and PQ = 9x - 25,
find the length of PR
Try It!
If PR = 3x + 5
and PQ = 9x - 25,
find the length of PR
PR = PQ
3x + 5 = 9x - 25
30 = 6x
5 = x

PR = 3x + 5
PR = 3(5) + 5
PR = 15 + 5
PR = 20
Now go to the class site
and take the tangents mini quiz
Full transcript