Download Circles and Chords

CIRCLES
AND
CHORDS
Equidistant – equally distant
Perpendicular Lines – intersect at right angles
Bisect – divide into two equal parts
Perpendicular Bisector – a line which bisects another line at right angles
Circle – the set of all points equidistant from a fixed central point
Circumference – the distance around a circle
Arc – part of the circumference of a circle
Chord – a line segment which joins two points on the circumference of a circle
Diameter – a chord which passes through the centre of a circle
Radius – a line segment which joins the centre of a circle to any point on the
circumference of the circle (the radius is half the length of the diameter)
Function – a relation for which every x-coordinate has one and only one y-coordinate
(vertical line test)
Converse – a statement formed by interchanging the subject and predicate
Statement: If p, then q.
Converse: If q, then p.
If a statement is true, its converse will not necessarily be true.
Example:
If it’s raining, then the grass is wet.
If the grass is wet, then it’s raining.
If a statement and its converse are both true then the statement can be written
using “iff” (if and only if)
Example:
If a shape has three sides, then it’s a triangle.
If a shape is a triangle, then it has three sides.
A shape is a triangle iff it has three sides.
CHORD PROPERTIES:
 Two chords are equidistant from the centre of a circle if and only if the chords are
of equal length.
 If a line bisects a chord and the line is perpendicular to that chord, then the line
passes through the centre of the circle.
 If a line is perpendicular to a chord and passes through the centre of the circle, then
the line bisects that chord.
 If a line passes through the centre of a circle and bisects a chord of that circle, then
the line is perpendicular to that chord.
CONGRUENCE POSTULATES
Congruent – same in size and shape
Side-Side-Side (SSS)
If the three sides of one triangle are congruent to the three sides of another
triangle, then the triangles are congruent.
Side-Angle-Side (SAS)
If two sides and the contained angle of one triangle are congruent to two sides and
the contained angle of another triangle, then the triangles are congruent.
Angle-Side-Angle (ASA)
If two angles and the contained side of one triangle are congruent to two angles
and the contained side of another triangle, then the triangles are congruent.
Side-Angle-Angle (SAA)
If two angles and a non-contained side of one triangle are congruent to two angles
and a non-contained side of another triangle, then the triangles are congruent.
Hypotenuse-Leg (HL)
If the hypotenuse and a leg of one triangle are congruent to the hypotenuse and a
leg of another triangle, then the triangles are congruent.
ANGLE THEOREMS
Vertical Angles – two nonadjacent angles formed by two intersecting lines. Vertical
angles are congruent. In the diagram below, 1 and 3 are vertical angles, and 2
and 4 are vertical angles.
4
1
3
2
Corresponding Angles – two angles occupying corresponding positions when two lines are
intersected by a transversal. Corresponding angles are congruent if the two lines
are parallel. In the diagram below, 1  5, 2  6, 3  7, and 4  8.
1 2
3 4
5 6
7 8
Alternate Interior Angles – two angles lying between two lines and on opposite sides of
the transversal. Alternate interior angles are congruent if the two lines are
parallel. In the diagram above, 3  6 and 4  5.
FORMULAS
Area of a Circle = πr2
Circumference of a Circle = 2πr or πd
Pythagorean Theorem: hyp2 = a2 + b2
Distance: d =
(x 2  x1 )2  (y 2  y1 )2
 x1  x2 y1  y2 
,

2 
 2
Midpoint: M 
Slope: m=
y 2  y1
x 2  x1
The slopes of two lines are equal iff the lines are parallel.
The slopes of two lines are negative reciprocals of each other
iff the lines are perpendicular.
ANGLES & ARCS
Minor arc – an arc that is smaller than a semi-circle; it is named using
its two endpoints and may also include a point on the arc (AB or ADB)
Magor arc – an arc that is larger than a semi-circle; it is named
using its two endpoints plus a point on the arc (BFA or BEA)
E
A
D
C
F
Central angle – an angle that has its vertex at the centre of a circle
(ACB)
B
Inscribed angle – an angle that has its vertex on the circle and has arms that contain
chords of the circle (AFB)
Subtend/Intercept – AB subtends ACB; ACB intercepts AB
Measure of an arc – measure of the central angle subtended by the arc (mAB = mACB)
PROPERTIES
OF
INSCRIBED ANGLES
The measure of an inscribed angle is equal to half the measure of the corresponding
central angle subtended by the same arc.
The measure of an arc is equal to double the measure of the inscribed angle it subtends.
Inscribed angles subtended by the same arc are congruent.
An angle inscribed in a semi-circle is a right angle.
Opposite angles in a cyclic quadrilateral are supplementary.
TANGENT PROPERTIES
1.
A tangent line to a circle is perpendicular to the radius drawn to the point of
tangency.
2.
A line drawn perpendicular to a tangent on a circle at the point of tangency passes
through the centre of the circle.
3.
Tangents to a circle drawn from any common external point are congruent.
4.
The angle between a tangent and a chord is congruent to the inscribed angle on the
opposite side of the chord.
TRANSFORMING CIRCLES
The standard form for the equation of a circle with centre at the origin (0,0) and
radius, r, is x2 + y2 = r2.
If a circle has its centre at the origin and a radius of 1, then the equation is x2 + y2 = 1,
and it’s called a unit circle. All other circles are compared to the unit circle.
If a circle does not have its centre at the origin, but at apoint (h, k) instead, then the
standard equation is (x – h)2 + (y – k)2 = r2.
Compared to the unit circle, its centre is shifted h units horizontally and k units
vertically, and its radius is r times larger or smaller. These transformations can be
summarized in mapping notation: (x, y)  (rx + h, ry + k)