Archive for conic section

It’s Maths!!

Posted in General with tags , , on January 20, 2010 by Admin

Maths Forum

Not only talks about Conic section(Circle) but also

other math topics.

KLIK to view;)

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Here’s another link which I think helps a lot for those who had troubles with Conic Section

Click HERE or HERE

or maybe HERE.

The Tab Tutor program sure is useful;)

 

moderator,

Nor Hidayah Binti Kamin

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Circle Theorems

Posted in The Angle of Circle with tags , , , , , , , , , , , , on January 10, 2010 by merrusmsk

Circle Theorems

Circles

A circle is a set of points which are all a certain distance from a fixed point known as the centre.
A line joining the centre of a circle to any of the points on the circle is known as a radius.

The circumference of a circle is the length of the circle. The circumference of a circle = 2 × π × the radius.

a sector, an arc and chord

The red line in the second diagram is called a chord. It divides the circle into a major segment and a minor segment.

Theorems

Angles Subtended on the Same Arc

Angles subtended on the same arc

Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.

Angle in a Semi-Circle

angle in a semi-circle

Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. So c is a right angle.

This proof is higher tier Proof

We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.

Divide the triangle in two

We know that each of the lines which is a radius of the circle (the green lines) are the same length. Therefore each of the two triangles is isosceles and has a pair of equal angles.

Two isosceles triangles

But all of these angles together must add up to 180°, since they are the angles of the original big triangle.

Therefore x + y + x + y = 180, in other words 2(x + y) = 180.
and so x + y = 90. But x + y is the size of the angle we wanted to find.

Tangents

A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle- it just touches it).

A tangent to a circle forms a right angle with the circle’s radius, at the point of contact of the tangent.

angle with a tangent

Also, if two tangents are drawn on a circle and they cross, the lengths of the two tangents (from the point where they touch the circle to the point where they cross) will be the same.

Tangents from an external point are equal in length

Angle at the Centre

Angle at the centre

The angle formed at the centre of the circle by lines originating from two points on the circle’s circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. a = 2b.

This proof is higher tier Proof

You might have to be able to prove this fact:

proof diagram 1

OA = OX since both of these are equal to the radius of the circle. The triangle AOX is therefore isosceles and so ∠OXA = a
Similarly, ∠OXB = b

proof diagram 2
Since the angles in a triangle add up to 180, we know that ∠XOA = 180 – 2a
Similarly, ∠BOX = 180 – 2b
Since the angles around a point add up to 360, we have that ∠AOB = 360 – ∠XOA – ∠BOX
= 360 – (180 – 2a) – (180 – 2b)
= 2a + 2b = 2(a + b) = 2 ∠AXB

This section is higher tier Alternate Segment Theorem

Alternate segment theorem

This diagram shows the alternate segment theorem. In short, the red angles are equal to each other and the green angles are equal to each other.

Proof

You may have to be able to prove the alternate segment theorem:

proof of alternate segment theorem

We use facts about related angles:

A tangent makes an angle of 90 degrees with the radius of a circle, so we know that ∠OAC + x = 90.
The angle in a semi-circle is 90, so ∠BCA = 90.
The angles in a triangle add up to 180, so ∠BCA + ∠OAC + y = 180
Therefore 90 + ∠OAC + y = 180 and so ∠OAC + y = 90
But OAC + x = 90, so ∠OAC + x = ∠OAC + y
Hence x = y

Cyclic Quadrilaterals

cyclic quadrilateral is a four-sided figure in a circle, with each vertex (corner) of the quadrilateral touching the circumference of the circle. The opposite angles of such a quadrilateral add up to 180 degrees.

This proof is higher tier Area of Sector and Arc Length

A sector

If the radius of the circle is r,
Area of sector = πr2 × A/360
Arc length = 2πr × A/360

In other words, area of sector = area of circle × A/360
arc length = circumference of circle × A/360

Fof more detail just type circle theorem in ur search list..

by

Melson Manggis