6.1 circles and related segments and angles circle - set of all points in a plane that are a fixed...
TRANSCRIPT
6.1 Circles and Related Segments and Angles
• Circle - set of all points in a plane that are a fixed distance from a given point (center).
• Radius – segment from the center to the edge of the circle. All radii of a circle are the same length.
• Concentric circles - circles that have the same center.
r
6.1 Circles and Related Segments and Angles
• Definition: chord – segment that joins 2 points of the circle.
• Theorem: A radius that is perpendicular to a chord bisects the chord.
6.1 Circles and Related Segments and Angles
• Definition: Arc – is a part of a circle that connects points A and B– Minor arc (<180)– Semicircle (180)– Major arc (> 180)
• Sum of all the arcs in a circle = 360
AB
6.1 Circles and Related Segments and Angles
• Central angle - vertex is the center of the circle and sides are the radii of the circle.
6.1 Circles and Related Segments and Angles
• Postulate 16:
Congruent arcs – arcs with equal measure.• Postulate 17: (Arc-addition postulate) If B lies
between A and C, then
A
B
C
)()(BCmBACm
)()()(ABCmBCmABm
6.1 Circles and Related Segments and Angles
• Inscribed angle – vertex is a point on the circle and the sides are chords of the circle.
• The measure of an inscribed angle is ½ the measure of its intercepted arc.
C
A
B
)()( 21
ABmACBm
6.1 Circles and Related Segments and Angles
• Example: Given circle with center O,
What is mAOB, mACB,
and
C
O
A
B
130)( and 80)(ACmABm
?)(CBm
6.2 More Angle Measures in a Circle
• Tangent - a line that intersects a circle at exactly one point.
• Secant - a line, segment, or ray that intersects a circle at exactly 2 points.
6.2 Inscribed/Circumscribed Polygons
• Polygon inscribed in a circle - its vertices are points on the circle.Note: The circle is circumscribed about the polygon.
• Polygon circumscribed about a circle - all sides of the polygon are segments tangent to the circle.Note: The circle is inscribed in the polygon.
6.2 More Angle Measures in a Circle
• Given an angle formed by intersecting chords:
note: 1 is not a central angle because E is not the center
A
B
DE
C
1
BCmADmm 2
11
6.2 More Angle Measures in a Circle
• Given an angle formed by intersecting secants:
B
A
C
ED
BDmCEmAm 2
1
6.2 More Angle Measures in a Circle
• Given an angle formed by intersecting tangents:
B
A
C
D
BCmBDCmAm 2
1
6.2 More Angle Measures in a Circle
• Given an angle formed by an intersecting secant and tangent:
A
D
B
C
BCmBDmAm
2
1
6.2 More Angle Measures in a Circle
• The radius drawn to a tangent at the point of tangency is perpendicular to the tangent.
• Given a line tangent to a circle and a chord intersecting at the point of tangency: B
A C
ABmBACm 2
1
6.3 Line and Segment Relationships in the Circle
• Given an angle formed by intersecting chords:
A
B
DE
C
1
EDBEECAE
6.3 Line and Segment Relationships in the Circle
• Given an angle formed by intersecting secants:
B
A
C
ED
AEADACAB
6.3 Line and Segment Relationships in the Circle
• Given an angle formed by intersecting tangents:
B
A
C
D
ACAB
6.3 Line and Segment Relationships in the Circle
• Given an angle formed by an intersecting secant and tangent:
A
D
B
C
ADACAB 2
6.4 Inequalities for the circle
•
B
D
C
A
1 2
)()(21CDmABmmm
6.4 Inequalities for the circle
• Given a circle with center O and distances to the center are OP and OQ
A BC
D
O
P
Q
OQOPCDAB
6.4 Inequalities for the circle
• Given two chords in a circle:
A BC
D
)()(CDmABmCDAB
6.4 Summary: Inequalities for the circle
Chords Central Angles
Distances to Center
Arcs
AB < CD mAOB < mCOD
PO>QO
AB > CD mAOB > mCOD
PO<QO
A B
C
D
O
P
Q
)()(CDmABm
)()(CDmABm
6.5 Locus of Points
• A locus of points - set of all points that satisfy a given condition.– Example: Locus of points that are a fixed
distance (r) from a point (p) – a circle with center p and radius r
– Example: Locus of points that are equidistant from 2 fixed points – perpendicular bisector of the segment between them.
6.5 Locus of Points
• More Examples:– Locus of points in a plane that are equidistant
from the sides of an angle – the ray that bisects the angle.
– Locus of points in space that are equidistant from two parallel planes – a plane parallel to both planes but midway in between.
– Locus of all vertices of right triangles having a hypotenuse AB – circle of diameter AB.
6.6 Concurrence of Lines
• Concurrent - A number of lines are concurrent if they have exactly one point in common.
• The 3 angle bisectors of the angles of a triangle are concurrent.
• The 3 perpendicular bisectors of the sides of a triangle are concurrent.
• The 3 altitudes of the sides of a triangle are concurrent. (point of concurrence is called the orthocenter)
6.6 Concurrence of Lines
• The 3 medians of a triangle are concurrent at a point that is 2/3 the distance from any vertex to the midpoint of the opposite side.
• Definition:
– Centroid: point of concurrence of the 3 medians of a triangle
7.1 Areas and Initial Postulates
• Postulate 21 – Area of a rectangle = length width
w
l
7.1 Areas and Initial Postulates
• Area of a triangle = ½ base height
h
b
7.1 Areas and Initial Postulates
• Area of a parallelogram = base height
h
b
h b
7.1 Areas and Initial Postulates
Polygon Area
Square s2
Rectangle l w
Parallelogram b h
Triangle ½ (b h)
7.2 Perimeters and Areas of Polygons
Polygon Perimeter
Triangle a + b + c (3 sides)
Quadrilateral a + b + c + d (4 sides)
Polygon Sum of all lengths of sides
Regular Polygon #sides length of a side
7.2 Perimeters and Areas of Polygons
• Heron’s formula – If 3 sides of a triangle have lengths a, b, and c, then the area A of a triangle is given by:
• Why use Heron’s formula instead of A = ½ bh?
)(perimeter -semi theis s
where))()((
21 cbas
csbsassA
7.2 Perimeters and Areas of Polygons
• Area of a trapezoid – A = ½ h (b1 + b2)
h
b1
b2
7.2 Perimeters and Areas of Polygons
• Area of a kite (or rhombus) with diagonals of lengths d1 and d2 is given by A = ½ d1 d2
d1
d2
7.2 Perimeters and Areas of Polygons
• The ratio of the areas of two similar triangles equals the square of the ratio of the lengths of any two corresponding sides.
a1
A2
a2A1
2
2
1
2
1
a
a
A
A
7.3 Regular Polygons and Area
• Center of a regular polygon – common center for the inscribed and circumscribed circles.
• Radius of the regular polygon – joins the center of the regular polygon to any one of the vertices.
Radius
Center
7.3 Regular Polygons and Area• Apothem – is a line segment from the center to
one of the sides and perpendicular to that side.• Central angle – is an angle formed by 2
consecutive radii of the regular polygon.• Measure of the central angle:
apothem
central anglen
C
360
7.3 Regular Polygons and Area• Area of a regular polygon with apothem
length a and perimeter P is:
where P = number of sides length of side
apothem
s
aPA 21
nsP
7.4 Circumference and Area of a Circle
• Circumference of a circle:C = d = 2r 22/7 or 3.14
• Length of an arc – the length l of an arc whose degree measure is m is given by:
Cm
l
360
rm
l
7.4 Circumference and Area of a Circle
• Limits: Largest possible chorddiameter
• For regular polygons ( ) as n – apothem (a) r (radius)– perimeter (P) C(circumference) = 2r
• Area of a circle –
aPA 21
221 )2( rrrA
7.5 More Area Relationships in the Circle
• Sector – region in a circle bounded by 2 radii and their intercepted arc
2sec 360360
rm
Am
A circletor
rm
sector
7.5 More Area Relationships in the Circle
• Segment of a circle – region bounded by a chord and its minor/major arcNote: For problems use:
torsegmenttriangle AAA sec
7.5 More Area Relationships in the Circle
• Theorem: Area of a triangle = ½ rPwhere P = perimeter of the triangleand r = radius of the inscribed triangleNote: This can be used to find the radius of the inscribed circle.