10-1 circles &...
TRANSCRIPT
10-1 Circles & Circumference
Circle-
Two circles are congruent if and only if they have congruent radii
All circles are similar
Concentric circles are coplanar circles that have the same center
2 points of intersection
1 point of intersection
NO points of intersection
Radius- Formula- Chord- Diameter- Formula-
Circumference-
Formula-
11-3 (part1) Areas of Circles Area of a circle-
Example 4 Find the circumference of a helipad on the top of a hospital if the diameter of the pad is 37 meters.
Example 5 Find the diameter and radius of a circle to the nearest hundredth if the circumference of the circle is 842.5 millimeters.
Example 1 What is the area of the circular putting green shown to the nearest square foot?
Example 2 Use the Area of a Circle to Find a Missing Measure Find the radius of a circle with an area of 46
square centimeters.
11-3 (part2) Sectors Sector of a circle-
Area of a Sector
Example 1) A circular pizza has a diameter of 16 inches and is cut into 10 congruent slices. What is the area of one slice to the nearest hundredth?
Find the area of each shaded sector. Round to the nearest tenth.
ex 2) ex3) ex4) Ex5) Jason wants to make a spinner for a new board game he invented. The
spinner is a circle divided into 8 congruent pieces, what is the area of each piece to the nearest tenth?
10-2 Arcs and Chords TERM: DEFINE: DRAW and LABEL Arc
Central Angle
Minor Arc “shade” minor arc: LY
Semi-Circle “shade” semi-circle: ACB
Major Arc “shade” major arc: LUY Arc Addition Postulate Congruent arcs
1. Using the letters shown in the diagram, name: a. four central angles _____, _____, _____, _____
b. two semicircles _____, _____
c. four minor arcs _____, _____, _____, _____
d. four major arcs _____, _____, _____, _____
2. Using the same circle, find the measure of each arc.
a. _____mBC b. _____mAB c. _____mABC
d. _____mCD e. _____mABD f. _____mBDA
Example 3 Refer to the circle graph. a. Find the measurement of the central
angle for each category. b. Use the categories to identify any arcs that are congruent.
Example 4 Find each measure in F. Example 5 Find the length of ZY . Round to the nearest hundredth.
a. b.
Example 1 Find measures of Central Angles
Find mQAM.
Example 2
In O, mAOB = 35, mDOE = 42, and .
Identify each arc as a major arc, minor arc, or semicircle. Then find its measure.
a. m AB
b. mCDE
c. m AE
OB OC
A
B
F
E
C
D
45
85 X 80
Y
Z
6 in.
X 120
12 cm
Y Z
10-3 Arcs and Chords ARC-CHORD Relationships
Congruent central angles have congruent chords
Congruent chords have congruent arcs
Congruent arcs have congruent central angles
DIAMETER(radius)-CHORD Relationships
In a circle, if a radius (or diam.) is to a chord, then it bisects the chord and its arc
In a circle, the bisector of a chord is a radius (or diam.)
Example 3 Find NP. Example 4 Find QR.
In the same circle or in circles, 2 chords are if and only if they are equidistant from the center
Example 1 In the figures, J K and
MN PQ . Find PQ.
M
N
J
3x + 4
P Q
K
x + 12
Example 2 In S, m PQR = 86. Find m PQ .
P
Q
R
S
8 T
Example 5 In ⨀P, QR = 7x – 20 and TS = 3x. What is x?
10-4 Inscribed Angles
Inscribed Angle: Intercepted Arc: Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc.
Theorem 10.8 Theorem 10.9 An inscribed angle intercepts a semi-circle (or diam.), If a quadrilateral is inscribed in a circle, then if and only if the angle is a right angle. its opposite angles are supplementary.
BC is a diameter. The sum of the angles of a quadrilateral is m BAC D and are supplementary.
mBAC C and are supplementary.
If inscribed angles of a circle intercept the same arc, then the angles are congruent.
A
B C
Quadrilateral ABCD is inscribed in the circle. The circle is circumscribed.
Find the measure of each angle or segment for each figure.
1) mPS 3) mADC 2) m PRU 4) m DAE 5) x= 7) x= 6) m WUT 8) m Y 9) m R 11) x 10) x 12) m B
13) mRUS 14) a 15) m RUS
10-5 Tangents Tangent-
Example 1: Draw the common tangent(s)
Example 3: JK is tangent to H . Find x.
Example 2
1. the center _____
2. diameter _____
3. a point of tangency _____
4. three radii _____, _____, _____
5. a tangent _____
6. two chords _____, _____
Example 4: GH and KH are tangent to F . Find a. Example 5: RS and RT are tangent to Q . Find n.
Common tangent
Thm 10.10: In a plane, a line is tangent to a circle if and only if it is ______________ to a radius drawn to the point of tangency.
Thm 10.11: If 2 segment from the same exterior point are tangent to a circle then they are _____________.
Ex 7: Find x and the perimeter of the triangle. In the picture, the circle is inscribed and the triangle is circumscribed.
a. b.
Example 6:Find x if ZB is tangent to Y .
10-6 Secants, Tangents, and Angle Measures Secant-
Example 1 Find x.
a.
b.
c.
Interior Angles formed by 2 intersecting chords or secants
Formula:
A secant & a tangent intersect at a point of tangency
Formula:
C
A
DT
x
B
135
53
E
R
S
T
x
U
62 144 V
G
K H
L J
x
120
92
Example 2. Find each measure
a. mQPR
R
136
S
Q
P
b.m
F
D
82 C
E
Vertex of Angle PICTURE Angle Measure
On the circle
Inside the circle
Outside the circle
ex5) x= ex6) x= ex7) x=
10-7 Special Segments in a Circle CHORD-CHORD THM Formula:
SECANT-SECANT-THM Formula:
TANGENT-SECANT THM Formula:
Find x in each picture. ex1) ex2) ex3) ex4)
Find x in each picture. ex5) ex6)
Find x in each picture. ex7) ex 8)
ex9) ex10)
ex11) ex12)
10-8 Equations of Circles Equation of a Circle in Standard Form: Example 1 Write the equation of each circle. a. center at (1, -5), d = 6 b. center at (-6, 0), r = 8 Example 2 Write the equation of the circle with center at (–3, 5) that passes through (3, –3).
Example 3 The equation of a circle is x2 + y2 – 2x – 10y = –22. State the coordinates of the center and the measure of the radius. Then graph the equation.
Example 5 – Intersections with Circles
Find the point(s) of intersection between x2 + y2 = 20 and y = –2x.