Transcript Circle Geometry

Shape and Space
CIRCLE GEOMETRY
Circle Geometry
Rule 1 : ANGLE IN A SEMICIRCLE = 90°
A triangle drawn from the
two ends of a diameter will
always make an angle of
90° where it hits the edge
of the circle, no matter
where it hits.
a = 50°
40°
a
EXERCISES
What is the size of each angle? Copy each diagram.
a
45°
c
65°
63°
b
81°
d
Circle Geometry
Rule 2 : A TANGENT AND RADIUS MEET AT 90°
A tangent is a line that just
touches the edge of a
curve. If a tangent and
radius meet at the same
point, then the angle they
make is exactly 90°.
a = 30°
60°
a
EXERCISES
What is the size of each angle? Copy each diagram.
59°
a
c
47°
d
71°
b
33°
Circle Geometry
Rule 3 : TWO RADII CREATE AN ISOSCELES TRIANGLE
These triangles are
constructed using two radii.
Therefore they are isosceles
triangles.
a = 30°
a
EXERCISES
What is the size of each angle? Copy each diagram.
67°
a
c
220°
d
110°
b
Circle Geometry
Rule 4 : THE PERPENDICULAR CORD BISECTOR IS A DIAMETER
No matter where you draw
the cord, the line that cuts
it exactly in half at 90° will
go through the center of
the circle – the diameter
line.
a = 36°
a
EXERCISES
What is the size of each angle? Copy each diagram.
b
67°
a
d
c
41°
77°
Circle Geometry
Rule 5 : ANGLES IN THE SAME SEGMENT ARE EQUAL
All triangles drawn from a
chord will have the same
angle where they touch the
circle. Also the two angles
on opposite sides of the
chord add up to 180°
a
a
b
b
a
50°
a + b = 180°
b
a = 50°
b = 130°
EXERCISES
What is the size of each angle? Copy each diagram.
a
31°
43°
e
b
f
g
d
c
76°
h
Circle Geometry
Rule 6 : ANGLE AT THE CENTRE IS TWICE THE ANGLE AT THE EDGE
Within the same segment,
the angle made at the
centre of a circle is exactly
double the angle formed by
the same chord at the
edge of the circle.
a
a
a
2a
63°
a
a = 126°
EXERCISES
What is the size of each angle? Copy each diagram.
c
a
72°
d
b
79°
Circle Geometry
°
Rule 7 : OPPOSITE ANGLES OF A CYCLIC QUADRILATERAL ADDS TO 180°
A cyclic quadrilateral is a
four sided shape with
every corner touching the
circle. Both pairs of
opposite angles add up to
180°
a
b
c
d
a
b
a + c = 180°
b + d = 180°
87°
79°
a = 93°
b = 101°
EXERCISES
What is the size of each angle? Copy each diagram.
e
a
101°
111°
b
81°
89°
f
h
61°
72°
59°
c
d
Circle Geometry
Rule 8 : EQALITY OF TANGENTS FROM A POINT
The two tangents drawn
from an outside point
always touch the circle at
the same distance. This
creates a reflective
situation with two
congruent right angled
triangles.
a = 41°
b = 49°
c = 49°
41°
a
b
c
EXERCISES
What is the size of each angle? Copy each picture.
a
h i
b
c
d
g
31°
39°
j
e
l
f
24°
Circle Geometry
Rule 9 : ANGLE IN THE OPPOSITE SEGMENT IS EQUAL
The angle between the
tangent and a chord is
equal to the opposite angle
in the triangle in the other
segment. This is the most
difficult to remember!
a
a = 77°
b = 71°
c = 32°
b
a
b
71°
77°
a
c
b
EXERCISES
What is the size of each angle? Copy each picture.
b
a
68°
c
h
80°
89°
66°
68°
f
45°
91° d
e
a
c
b