Transcript Lesson 1 Contents

Lesson 10-R
Chapter 10 Review
Objectives
• Review Chapter 10 material
Parts of Circles
• Circumference (Perimeter)
– once around the outside of the circle;
Formulas: C = 2πr = dπ
• Chord
– segment with endpoints of the edge of the circle
• Radius
– segment with one endpoint at the center and one at the edge
• Diameter
– segment with endpoints on the edge and passes thru the center
– longest chord in a circle
– is twice the length of a radius
• Other parts
– Center: is also the name of the circle
– Secant: chord that extends beyond the edges of the circle
– Tangent: a line (segment) that touches the circle at only one point
Arcs in Circles
•
•
•
•
Arc is the edge of the circle between two points
An arc’s measure = measure of its central angle
All arcs (and central angles) have to sum to 360°
If two arcs have the same measure then the chords that form those
arcs have the same measure
• If a radius is perpendicular to a chord then it bisects the chord and
the arc formed by the chord (example arc AED below)
• Major Arc (example: arc DAB)
– measures more than 180°
– more than ½ way around the circle
BE is a diameter
and AB = AD
120°
• Minor Arc (example: arc AED)
– measures less than 180°
– less than ½ way around the circle
• Semi-circle (example: arc EAB)
– measures 180°
– defined by a diameter
B
A
C
120°
60°
E
60°
D
Angles Associated with Circles
Name
Vertex
Location
Sides
Formula
Example
Central
Center
radii
= measure of the arc
BCD = 110°
Inscribed
Edge
chords
= ½ measure of the arc
BAD = 55°
Interior
Inside
chords
= average of the vertical arcs
EVH = 73°
Exterior
Outside
Secants /
Tangents
= ½ (Big Arc – Little Arc)
= ½ (Far Arc – Near Arc)
NVM = 30°
minor arc LK = 10°
minor arc NM = 70°
minor arc FG = 110°
minor arc EH = 36°
minor arc BD = 110°
V
K
L
B
A
36°
C
110°
D
F
E
H
10°
V
C
110°
C
G
M
N
70°
Segments Inside/Outside of Circles
• Segments that intersect inside or outside the circle have the length
of their parts defined by:
Two Chords
Inside a Circle
Two Secants
From Outside Point
Secant & Tangent
From Outside Point
J
K
L
3
5
L
4
J
4
6
K
K
J
6
3
T
8
7
N
M
LJ · JM = NJ · JK
38=64
Inside the circle, it’s the
parts of the chords
multiplied together
9
11
N
M
M
JL · JN = JK · JM
5  12 = 4  15
JT · JT = JK · JM
6  6 = 3  12
Outside the circle, it’s the outside part
multiplied by the whole length
OW = OW
Tangents and Circles
• Tangents and radii always form a right angle
• We can use the converse of the Pythagorean theorem to check if a
segment is tangent
• The distance from a point outside the circle along its two tangents
to the circle is always the same distance
J
Example 1
Given:
JT is tangent to circle C
JC = 25 and JT = 20
S
T
Find the radius
JC² = JT² + TC²
25² = 20² + r²
625 = 400 + r²
225 = r²
15 = r
Example 2
Given:
same radius as example 1
JC = 25 and JS = 16
Is JS tangent to circle C?
C
JC² = JS² + SC²
25² = 16² + 15²
625 = 256 + 225
625 ≠ 481
JS is not tangent
Equation of Circles
• A circle’s algebraic equation is defined by:
(x – h)² + (y – k)² = r²
where
the point (h, k) is the location of the center of the circle
and r is the radius of the circle
• Circles are all points that are equidistant (that is the
distance of the radius) from a central point (the center)
Summary & Homework
• Summary:
–A
• Homework:
– study for the test