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,
$r$ , 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. (
$AB$ is a tangent to the circle at point
$P$ ).
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
$\u25b5ABC$ , this means that
${\left(AB\right)}^{2}+{\left(BC\right)}^{2}={\left(AC\right)}^{2}$
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
$O$ . Draw a chord
$AB$ and draw a perpendicular line from the centre of the circle to intersect the chord at point
$P$ .
The aim is to prove that
$AP$ =
$BP$
$\u25b5OAP$ and
$\u25b5OBP$ are right-angled triangles.
$OA=OB$ as both of these are radii and
$OP$ is common to both triangles.
Apply the Theorem of Pythagoras to each triangle, to get:
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
$O$ . Draw a chord
$AB$ and draw a line from the centre of the circle to bisect the chord at point
$P$ .
The aim is to prove that
$OP\perp AB$ In
$\u25b5OAP$ and
$\u25b5OBP$ ,
Theorem 3 The perpendicular bisector of a chord passes through the centre of the circle.
Proof :
Consider a circle. Draw a chord
$AB$ . Draw a line
$PQ$ perpendicular to
$AB$ such that
$PQ$ bisects
$AB$ at point
$P$ . Draw lines
$AQ$ and
$BQ$ .
The aim is to prove that
$Q$ is the centre of the circle, by showing that
$AQ=BQ$ . In
$\u25b5OAP$ and
$\u25b5OBP$ ,
$AP=PB$ (given)
$\angle QPA=\angle QPB$ (
$QP\perp AB$ )
$QP$ is common to both triangles.
$\therefore \u25b5QAP\equiv \u25b5QBP$ (SAS).
From this,
$QA=QB$ . 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,
$Q$ must be the centre of the circle.
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?