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
• 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
A
D
C 120°
BE is a diameterand AB = AD
B
E
120°
60°
60°
Angles Associated with Circles
NameVertex
LocationSides 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 OutsideSecants / Tangents
= ½ (Big Arc – Little Arc)= ½ (Far Arc – Near Arc)
NVM = 30°
V
KL
MN
A
D
B
C 110°
minor arc BD = 110°
E
G
F
C 110°H
36° V
minor arc FG = 110°minor arc EH = 36°
minor arc LK = 10°minor arc NM = 70°
C
70°
10°
Segments Inside/Outside of Circles• Segments that intersect inside or outside the circle have the length
of their parts defined by:
J
J
K
KL
M
M
N
J
K
T
M
LJ · JM = NJ · JK3 8 = 6 4 JL · JN = JK · JM
5 12 = 4 15JT · JT = JK · JM
6 6 = 3 12
Two ChordsInside a Circle
Two SecantsFrom Outside Point
Secant & Tangent From Outside Point
L
N
Inside the circle, it’s the parts of the chordsmultiplied together
Outside the circle, it’s the outside part multiplied by the whole length OW = OW
4
68
3
63
9
5 4
117
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
Example 1Given:JT is tangent to circle CJC = 25 and JT = 20
Find the radius
ST
J
C
Example 2Given:same radius as example 1JC = 25 and JS = 16
Is JS tangent to circle C?
JC² = JT² + TC²25² = 20² + r²625 = 400 + r²225 = r²15 = r
JC² = JS² + SC²25² = 16² + 15²625 = 256 + 225625 ≠ 481JS 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 circleand 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
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