by,

Kong Ngiik Hiong

Stuffs about Circle…MATHS!!

by,

Kong Ngiik Hiong

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An **inscribed angle** is an angle formed by two chords in a circle which have a common endpoint. This common endpoint forms the vertex of the inscribed angle. The other two endpoints define what we call an **intercepted arc** on the circle. The intercepted arc might be thought of as the part of the circle which is “inside” the inscribed angle. (See the pink part of the circle in the picture above.)

A **central angle** is any angle whose vertex is located at the center of a circle. A central angle necessarily passes through two points on the circle, which in turn divide the circle into two arcs: a **major arc** and a **minor arc**. The minor arc is the smaller of the two arcs, while the major arc is the bigger. We define the **arc angle** to be the measure of the central angle which intercepts it.

The **Inscribed Angle Conjecture I** gives the relationship between the measures of an inscribed angle and the intercepted arc angle. It says that the measure of the intercepted arc is **twice** that of the inscribed angle.

The precise statements of the conjectures are given below. Each conjecture has a linked Sketch Pad demonstration to illustrate its truth (proof by Geometer’s Sketch Pad!). The linked activities sheet also include directions for further “hands on” investigations involving these conjectures, as well as geometric problems which utilize their results.

**Conjecture ( Inscribed Angles Conjecture I ):** In a circle, the measure of an inscribed angle is half the measure of the central angle with the same intercepted arc..

**Proof:** The measure of each inscribed angle is exactly half the measure of its intercepted arc. Since they have the same intercepted arc, they have the same measure.

**Proof:** The intercepted arc for an angle inscribed in a semi-circle is 180 degrees. Therefore the measure of the angle must be half of 180, or 90 degrees. In other words, the angle is a right angle.

by,

Kong Ngiik Hiong

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.

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

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

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**.

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.

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.

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.

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.

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.

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**.

You might have to be able to prove this fact:

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

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 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.

You may have to be able to prove the 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

A **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.

If the radius of the circle is r,

Area of sector = πr^{2} × 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

In everyone’s experience it is usual to measure angles in degrees. We learn early in childhood that there are 360 degrees in a circle, that there are 90 degrees in a right angle, and that the angle of an equilateral triangle contains 60 degrees. On the other hand, to scientists, engineers, and mathematicians it is usual to measure angles in radians.

The size of a radian is determined by the requirement that there are 2 radians in a circle. Thus 2 radians equals 360 degrees. This means that 1 radian = 180/ degrees, and 1 degree = /180 radians.

The reason for this is that so many formulas become much easier to write and to understand when radians are used to measure angles. A very good example is provided by the formula for the length of a circular arc. If `A` and `B` are two points on a circle of radius R and center C, then the length of the arc of the circle connecting them is given by

`d(A,B) = R a,`

where `R` is the radius of the sphere, and `a` is the angle `ACB` measured in radians. If we measure the angle in degrees, then the formula is

`d(A,B) = R a/180,`

These formulas can be checked by noticing that the arc length is proportional to the angle, and then checking the formula for the full circle, i.e., when a = 2 radians (or 360 degrees).

The figure below gives the relationship between degrees and radians for the most common angles in the unit circle measured in the counterclockwise direction from the point to the right of the vertex. The form of the ordered pair is {degree measure, radian measure}

In mathematics and physics, the **radian** is a unit of angle measure. It is the SI derived unit of angle. It is defined as the angle subtended at the center of a circle by an arc of circumference equal in length to the radius of the circle. Angle measures in radians are often given without any explicit unit. When a unit is given, it is **rad**.

There are 2 π (about 6.283185) radians in a complete circle, so:

2π rad = 360°

1 rad = 360/(2π)° = 180/π° (approximately 57.29578°).

or:

360° = 2π rad

1° = 2π/360 rad = π/180 rad

In calculus, angles must be represented in radians in trigonometric functions, to make identities and results as simple and natural as possible. For example, the use of radians leads to the simple identity

which is the basis of many elegant identities in mathematics.

The radian was formerly an SI supplementary unit, but this category was abolished from the SI system in 1995.

If the end points of an arc are joined to a third point on the circumference of a circle, then an angle is formed.

For example, the minor arc *AB* subtends an angle of 45º at *C*. The angle *ACB* is said to be the angle subtended by the minor arc *AB* (or simply arc *AB*) at *C*.

The angle *ACB* is an angle at the circumference standing on the arc *AB*.

If the end points of an arc are joined to the centre of a circle, then an angle is formed. For example, the minor arc *AB* subtends an angle of 105º at *O*. The angle *AOB *is said to be the angle subtended by the minor arc *AB* (or simply arc *AB*) at the centre *O*.

The angle *AOB* is an angle at the centre *O *standing on the arc *AB*.

Use the information given in the diagram to prove that the angle at the centre of a circle is twice the angle at the circumference if both angles stand on the same arc.

The angle at the centre of a circle is twice the angle at the circumference if both angles stand on the same arc. This is called the **Angle at Centre Theorem**.

We also call this the **basic property**, as the other angle properties of a circle can be derived from it.

Find the value of the pronumeral in the following circle centred at *O*.

Find the value of the pronumeral in the following circle centred at *O*.

The angle in a semi-circle is a right angle.

Find the value of the pronumeral in the following circle centred at *O*.

by,

Kong Ngiik Hiong

The concept of angle is one of the most important concepts in geometry. The concepts of equality, sums, and differences of angles are important and used throughout geometry, but the subject of trigonometry is based on the *measurement* of angles.

There are two commonly used units of measurement for angles. The more familiar unit of measurement is that of degrees. A circle is divided into 360 equal degrees, so that a right angle is 90°. For the time being, we’ll only consider angles between 0° and 360°, but later, in the section on trigonometric functions, we’ll consider angles greater than 360° and negative angles.

Degrees may be further divided into minutes and seconds, but that division is not as universal as it used to be. Parts of a degree are now frequently referred to decimally. For instance seven and a half degrees is now usually written 7.5°. Each degree is divided into 60 equal parts called *minutes.* So seven and a half degrees can be called 7 degrees and 30 minutes, written 7° 30′. Each minute is further divided into 60 equal parts called *seconds,* and, for instance, 2 degrees 5 minutes 30 seconds is written 2° 5′ 30″. The division of degrees into minutes and seconds of angle is analogous to the division of hours into minutes and seconds of time.

Usually when a single angle is drawn on a *xy*-plane for analysis, we’ll draw it with the vertex at the origin (0,0), one side of the angle along the *x*-axis, and the other side above the *x*-axis.

The other common measurement for angles is radians. For this measurement, consider the unit circle (a circle of radius 1) whose center is the vertex of the angle in question. Then the angle cuts off an arc of the circle, and the length of that arc is the radian measure of the angle. It is easy to convert between degree measurement and radian measurement. The circumference of the entire circle is 2 ( is about 3.14159), so it follows that 360° equals 2 radians. Hence, 1° equals /180 radians, and 1 radian equals 180/ degrees.

Most calculators can be set to use angles measured with either degrees or radians. Be sure you know what mode your calculator is using.

Although the word “radian” was coined by Thomas Muir and/or James Thompson about 1870, mathematicians had been measuring angles that way for a long time. For instance, Leonhard Euler (1707-1783) in his *Elements of Algebra* explicitly said to measure angles by the length of the arc cut off in the unit circle. That was necessary to give his famous formula involving complex numbers that relates the sign and cosine functions to the exponential function *e ^{it}* = cos

where *t* is what was later called the radian measurment of the angle. Unfortunately, an explanation of this formula is well beyond the scope of these notes. But, for a little more information about complex numbers, see my Short Course on Complex Numbers.

An alternate definition of radians is sometimes given as a ratio. Instead of taking the unit circle with center at the vertex of the angle, take any circle with center at the vertex of the angle. Then the radian measure of the angle is the ratio of the length of the subtended arc to the radius of the circle. For instance, if the length of the arc is 3 and the radius of the circle is 2, then the radian measure is 1.5.

The reason that this definition works is that the length of the subtended arc is proportional to the radius of the circle. In particular, the definition in terms of a ratio gives the same figure as that given above using the unit circle. This alternate definition is more useful, however, since you can use it to relate lengths of arcs to angles. The formula for this relation is

radian measure times radius = arc length

For instance, an arc of 0.3 radians in a circle of radius 4 has length 0.3 times 4, that is, 1.2.

Below is a table of common angles in both degree measurement and radian measurement. Note that the radian measurement is given in terms of . It could, of course, be given decimally, but radian measurement often appears with a factor of .

Angle | Degrees | Radians |
---|---|---|

90° | /2 | |

60° | /3 | |

45° | /4 | |

30° | /6 |

by,

Kong Ngiik Hiong