Archive for the The Angle of Circle Category

Inscribed Angles

Posted in The Angle of Circle with tags , on January 20, 2010 by vipvvip

by,

Kong Ngiik Hiong

angle of circle

Posted in The Angle of Circle with tags on January 20, 2010 by vipvvip

Inscribed Angles Conjectures



Explanation:

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.


The precise statement of the conjectures:

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..


Corollary (Inscribed Angles Conjecture II ): In a circle, two inscribed angles with the same intercepted arc are congruent.

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.


Corollary (Inscribed Angles Conjecture III ): Any angle inscribed in a semi-circle is a right angle.

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

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

angle of circle

Posted in The Angle of Circle with tags , on December 25, 2009 by vipvvip

Degree/Radian Circle

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.

Angle at the Circumference

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.

In the circle, the angle subtended by the minor arc AB at C is 45 degrees.

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

Angle at the Centre

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 minor arc AB subtends the angle at the centre, AOB, of size 105 degrees.

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

Angle at Centre Theorem

Theorem

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.

In the circle, angle AOB at the centre is subtended by the arc AB and angle ACB at the circumference is also subtended by the arc AB.  The diameter of the circle passing through O and touching C forms angles of size a degrees and b degrees near the point C and x degrees and y degrees near the centre O.

Given:

Angle AOB and angle ACB stand on the same arc; and O is the centre of the circle.

To prove:

Angle AOB = 2(angle ACB)

Proof:

In the circle, angle AOB at the centre is subtended by the arc AB and angle ACB at the circumference is also subtended by the arc AB.  The diameter of the circle passing through O and touching C forms angles of size a degrees and b degrees near the point C and x degrees and y degrees near the centre O.  Angle CAO is a degrees and angle CBO is b degrees.

From triangle OAC, x = a + a  {Exterior angle of a triangle}.  So, x = 2a  ...(1).  From triangle OBC, y = b + b  {Exterior angle of a triangle}.  So, y = 2b  ...(2).  Adding (1) and (2) gives x + y = 2a + 2b = 2(a + b).  So, angle AOB = 2(angle ACB) as required.

In general:

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.

Example 22

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

In the circle, the angle AOB at the centre subtended by arc AB is 4x degrees and the angle ACB at the circumference subtended by arc AB is 60 degrees.

Solution:

Example 23

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

Solution:

Angle in a Semi-Circle

In general:

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

Example 24

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

Solution:

by,

Kong Ngiik Hiong

angles in circle

Posted in The Angle of Circle with tags on December 25, 2009 by wongswanswan

Formulas for Working with Angles in Circles
(Intercepted arcs are arcs “cut off” or “lying between” the sides of the specified angles.)

There are basically five circle formulas that
you need to remember:               
 
1.  Central Angle:  
A central angle is an angle formed by two intersecting radii such that its vertex is at the center of the circle.
 

Central Angle = Intercepted Arc

<AOB is a central angle.
 Its intercepted arc is the minor arc from A to B.  m<AOB = 80º
 

2.  Inscribed Angle:
An inscribed angle is an angle with its vertex “on” the circle, formed by two intersecting chords.

Inscribed Angle = Intercepted Arc

<ABC is an inscribed angle.
 Its intercepted arc is the minor arc from A to C.
m<ABC = 50º

          
3.  Tangent Chord Angle:
An angle formed by an intersecting tangent and chord has its vertex “on” the circle.

Tangent Chord Angle =
 Intercepted Arc

<ABC is an angle formed by a tangent and chord.
Its intercepted arc is the minor arc from A to B.
m<ABC = 60º
 

4.  Angle Formed Inside of a Circle by Two Intersecting Chords:
When two chords intersect “inside” a circle, four angles are formed.  At the point of intersection, two sets of vertical angles can be seen in the corners of the X that is formed on the picture.  Remember:  vertical angles are equal.

Angle Formed Inside by Two Chords =
 
Sum of Intercepted Arcs 
Once you have found ONE of these angles, you automatically know the sizes of the other three by using your knowledge of vertical angles (being equal) and adjacent angles forming a straight line (adding to 180).

 


<BED is formed by two intersecting chords.
 Its intercepted arcs are .
  [Note:  the intercepted arcs belong to the set of vertical angles.]
 
also, m<CEA = 120º (vetical angle)
m<BEC and m<DEA = 60º by straight line.
5.  Angle Formed Outside of a Circle by the Intersection of:
“Two Tangents” or “Two Secants” or “a Tangent and a Secant”.

The formulas for all THREE of these situations are the same:
Angle Formed Outside = Difference of Intercepted Arcs 
(When subtracting, start with the larger arc.)
Two Tangents:
<ABC is formed by two tangents intersecting outside of circle O. 
The intercepted arcs are minor arc AC and major arc AC.  These two arcs together comprise the entire circle.
Special situation for this set up:  It can be proven that <ABC and central <AOC are supplementary.  Thus the angle formed by the two tangents and its first intercepted arc also add to 180º.
Two Secants:
<ACE is formed by two secants intersecting outside of circle O. 
The intercepted arcs are minor arcs BD and AE. 
a Tangent and a Secant:
<ABD is formed by a tangent and a secant intersecting outside of circle O. 
The intercepted arcs are minor arcs AC and AD. 

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

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.

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

 by,

Wong Swan Swan

 

 

angle of circle

Posted in The Angle of Circle with tags , on December 25, 2009 by vipvvip

Angle measurement

The concept of angle

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.

angles: 15, 30, 45 degrees

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&deg. 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.

unit circle with angle in standard position

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 2pi (pi is about 3.14159), so it follows that 360° equals 2pi radians. Hence, 1° equals pi/180 radians, and 1 radian equals 180/pi 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.

Short note on the history of radians

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 eit = cos t + i sin t

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.

Radians and arc length

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 pi. It could, of course, be given decimally, but radian measurement often appears with a factor of pi.

Angle Degrees Radians
90 degree angle 90° pi/2
60 degree angle 60° pi/3
45 degree angle 45° pi/4
30 degree angle 30° pi/6

 

by,

Kong Ngiik Hiong

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