Research one of the following geometrical ideas and describe it to your group:
taxicab geometry,
spherical geometry,
fractals,
the Koch snowflake.
Circle geometry
Terminology
The following is a recap of terms that are regularly used when referring to circles.
An arc is a part of the circumference of a circle.
A chord is defined as a straight line joining the ends of an arc.
The radius,
, is the distance from the centre of the circle to any point on the circumference.
The diameter is a special chord that passes through the centre of the circle. The diameter is the straight line from a point on the circumference to another point on the circumference, that passes through the centre of the circle.
A segment is the part of the circle that is cut off by a chord. A chord divides a circle into two segments.
A tangent is a line that makes contact with a circle at one point on the circumference. (
is a tangent to the circle at point
).
Axioms
An axiom is an established or accepted principle. For this section, the following are accepted as axioms.
The Theorem of Pythagoras, which states that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. In
, this means that
A tangent is perpendicular to the radius, drawn at the point of contact with the circle.
Theorems of the geometry of circles
A theorem is a general proposition that is not self-evident but is proved by reasoning (these proofs need not be learned for examination purposes).
Theorem 1 The line drawn from the centre of a circle, perpendicular to a chord, bisects the chord.
Proof :
Consider a circle, with centre
. Draw a chord
and draw a perpendicular line from the centre of the circle to intersect the chord at point
.
The aim is to prove that
=
and
are right-angled triangles.
as both of these are radii and
is common to both triangles.
Apply the Theorem of Pythagoras to each triangle, to get:
However,
. So,
This means that
bisects
.
Theorem 2 The line drawn from the centre of a circle, that bisects a chord, is perpendicular to the chord.
Proof :
Consider a circle, with centre
. Draw a chord
and draw a line from the centre of the circle to bisect the chord at point
.
The aim is to prove that
In
and
,
(given)
(radii)
is common to both triangles.
(SSS).
Theorem 3 The perpendicular bisector of a chord passes through the centre of the circle.
Proof :
Consider a circle. Draw a chord
. Draw a line
perpendicular to
such that
bisects
at point
. Draw lines
and
.
The aim is to prove that
is the centre of the circle, by showing that
. In
and
,
(given)
(
)
is common to both triangles.
(SAS).
From this,
. Since the centre of a circle is the only point inside a circle that has points on the circumference at an equal distance from it,
must be the centre of the circle.
Questions & Answers
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature