Mathematics : The angle at centre is two times to the angle at circumference

Name of School:

Mathematics Lesson Plane No .5 Inhlakanipho High

Student Surname and Name: Student Number:

Maphumulo Mthokozisi 207503878

Grade:

10 and 11

Subject:

Mathematics

Topic: Content /Concept Area:

The angle subtended by an arc or chord Circle, circumference and subtended angles and

Caps Page No. Duration of Lesson: Date :

chord.

173 45 minutes 15 / o6 / 2016

The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre);

Specific Aims: To investigate and prove the theorems of the geometry of circles.

To develop and revive the cognitive knowledge of the circles for advancement of the upcoming geometric conjectures.

To explore properties of circle in advance.

Knowledge Learners should gain

Skills Learners should be able to

Values and Attitudes Learners should acquire

knowledge and

do following :

values and attitudes

understanding of the following: Recall the factors of a circle

conducive to : Draw and label the

Produce a drawing of a

properties of a circle to

circle in which a chord

identify a chord.

subtends angle in the circle

Explain the angles at the

Identify angle at centre and

(circumference of a circle).

Devise ways to interpret the

circumference and those at

the angle at the

angles subtended by an arc

arch Discuss properties of

circumference.

Analyse the exterior angles

at the centre of a circle.

Produce a completed the set

isosceles triangle and

of triangles from the line

of two angles at centre of a

triangle in particular.

drawn and produced with

circle = two angles in their

centre of a circle and

circumference of a circle.

Formulate the deductive

circumference Construct , transform and

Acknowledge the scientific

reasoning behind the why

analyse the angles at centre

calculations and cognitive

the angle at centre of a circle of a circle stating clear why

theory developed from

doubles the angle at the

they are also equal to the

schematic drawing aimed at

circumference

angle at the circumference

illustrating the relationship of the angles subtended by the chord within the circle in generally.

Bloom’s Taxonomy Remember, Understand, apply, analyse, evaluate, and create.

Approaches / Teaching Strategies:

Role plays.

Case studies Class discussions.

Questioning.

Whole class-discussion Group Work

Resources:

OHP/ Transparency Calculators Chalkboard Newspaper Worksheet Lesson Phases:

Introduction: During the introduction I will question students to explain what the circle is. The aim is to let them to recall different geometric properties of circle and geometric shape that may develop from further application of drawing. Then I will explain the main important concept and content regarding the topic that learners must deal with. In this classroom exercise we are going to consider a circle with a chord which subtends two angles in the circumference of a circle. We make use of Geo-gebra to analyse the relationship between angles at centre of a circle and the angles in the circumference of a circle. Briefly outline the properties of three main types of triangles and special lines that transform the circle and thereby enable us to interpret relations of angles within the circle. Hint we shall require the inscribed triangles and inscribed quadrilaterals. In the introduction we make use of various drawings to illustrate various angles and geometric shapes that influence relationship between angles subtended by chords. The aim of doing all of these things is to draw students’ attention to recall the basic geometric facts in line with the development of theory in question. Indeed the major task of this lesson plan is to prove that the angles at centre of a circle is twice the angles at the circumference of the circles.

IMPORTANT PROPERTIES OF CIRCLE

Segment

Chord

Diameter

Radius

Tangent

A diameter is a line that bisects the circle into two equal halves. A diameter starts from the circumference of the circle to pass through the centre of the circle and to the circumference again.

However, because the circle maintains equal distance from centre of the circle to the circumference one could draw a conclusion that diameter = 2 radii. Radius is a point on the circle of the circle, which starts from the circumference of the circle to the centre or vice versa.

A Chord: if you draw a line anywhere to join into two separate points of a circle except the circumcircle, that line is called a chord. A chord lies inside of the circle. A chord is the focus of this lesson, since it is the line that seem to subtend equal angles in the circumference. In other hand diameter, also act as a chord of some kind, since it is capable of creating a smooth relationship on which two right angles within two separate triangles are equal. In the above diagram line CD is a chord, EF is a diameter, AG is a radius. The space between chord and radian circumference as you move out of the circle is called segment.

The focus of this classroom exercise is to give a proof that an angle subtended by arc at the centre is two times or twice the angle at the circumference of the circle. Please notice that geometry theories overlap with each other. As a professional educator, I will introduce students to various rules mediating to the operation of an inscribed quadrilateral, inscribed triangles and some basic geometric figures including. The cases that discuss the fact that a chord subtends equal angles will not be proved in this lesson but , where they offer exciting case that overlap with an assumption that an angle at centre of a circle is twice the angle at circumference they may be discussed .

Development: It is important to use visual interactives for demonstration of geometric theories. Geometric theories are becoming important for making estimations. Algebra is becoming hectic about geometry, as you should have noticed that there are more number of challenges involving trigonometry where algebraic expressions look to integrate geometry and algebraic expressions. Leaning these theories thoroughly will offer kids insights to develop cognitive thinking of their independent decision making when dealing with geometry, algebra and trigonometry altogether.

This classroom exercise is watching to prove a theory, which states that an angle at the centre of a circle is twice the angle at the circumference of the circle.

Methods: i)

Draw a line from the circumference of the circle to pass through the centre of the circle so that two sets of different isosceles triangles are produced.

ii)

Do not take a complicated triangle; you will learn to do so as the time goes by. So consider taking an inscribed right angle triangle or a triangle on which one chord subtends one angle in the circumference of a circle plus two sets of radii.

iii)

Do not apply two triangles simultaneously because this will divert focus to appear as if you are looking at a theory that state that a chord subtends equal angles( know what you are doing ).

iv)

Derive the external angles of triangles to be equal into their two interior opposite angles separately.

v)

Give a statement for the above fact: teacher must not give student answers but they must discover themselves so that they are capable of analysing their conjectures in future. Keep in your mind that the isosceles triangle has two equal angles, which must help you to moderate your students.

vi)

State those three main reasons for why the angle at centre of the circle would be twice the angle at the circumference Reason: a teacher is not expected to reveal these reasons to students, but students must discover themselves, in form of delusive reasoning, recalling all work that they are supposed to have done in their grade 8. (Simple speaking the students are revising and learning through participation, their problem solving ability is being tested in their groups).

Reasons a) An exterior angle of a triangle is equal to two interior angle of a triangle except the one it supplements with the straight line. b) The triangle that was formed by produced dotted line was isosceles triangle so the two angles are crashed to take one expression (e.g. : 2 Y ° ) . Please try where possible to give a chance for students to interact with geogebra or use instruments. c) Combined set of angles at centre are now being factorised: when you take common factor of 2 you will notice that the two set of angles at the circumference of circle are left across the equal sign. Skilful students would be able to set the angle at the circumference appropriately so that it appear to be equal to the half of the angle at the centre of a circle. d) A teacher must consolidate the class lesson, students get feedback and errors are rectified.

Questions asked: Define what is circumference of a circle (hint perimeter)?

What is isosceles triangle and how would you identify it from other triangles?

Discuss what is the effect of drawing the dotted line produced from circumference of the circle of to the centre of the circle.

In your own words write down reasons why the exterior angle of the circle is equal to the interior opposite angles. (Use any form of evidence remembered) Describe reasons why the right-angled triangle subtended by the diameter would still appear under the theory, which asserts that the angle at centre of a circle is twice the angles at the circumference of a circle.

NB you should be informed that sometimes angles are given where you must make use of this theory to determine their actual calculation (future activity).

Consolidation

Below is a proof that line angle BAC = Two times the angle BDC

ȊȊȊ Statement: Angle A2 = angle D2 + angle DBA

Reason : -An exterior angle of a triangle = 2 x the interior opp. Angles.

But angle D2 = angle DBA

ADB is an isosceles triangle

Therefore Angle A2= D2 + D2

Symposium of the two similar angles

Angle A2= 2( D2) Similarly in triangle ADC Angle A1 = D1 + angle DCA

An exterior angle of a triangle = 2 x the interior opp. Angles.

D1 = angle DCA , = D1

∆ DCA is an isosceles triangle, where radii are equal; therefore, base angles must be equal.

Therefore angle A1 = 2( D1 + D1) Therefore A1 = 2( D1)

Symposium of the two angles due to above statement

Now combine the two angles and evaluate the results as follows Angle A1 + angle A2 = 2( D1) + 2( D2)

Integration of triangle statement: an exterior angle of a triangle = two interior opp. angles.

Angle A1 + angle A2 = 2( D1 + D2)

Factorization

But, Angle A1 + angle A2 = angle BAC

Please notice that this is the size of the angle in the centre of the circle

But also ,angles D1 + D2 = angle BDC , the angle in the circumference of the circle This implies that angle BAC, the angle at centre of the circle = twice the angle BDC, the angle at the circumference of the circle.

Homework: Take home homework

Question 1

Conclusion: the angle at centre of the circle is twice the angle at the circumference of the circle.

In the given circle above, A is the centre of a circle. BAC is a straight-line diameter and angle A 1 and angle A2 are complementary angle of a straight line. Using the theory that states if the angle at centre of the circle is two times to the angle at the circumference, prove that angle BAC = 2 (BDC) . (Hint A1 and A2 are exterior angles to D1 and D2 including their tween angles. Trey where possible to challenge the idea that if BAC is a straight line then angle BDC is a right angle.

Question 2: With the aid of a teacher, draw an inscribed circle so that you keep two supplementary angles. Ignore the numeric size of an angle but join the two radii to the centre of the circle-using dotted line or a plan of your choice.

Why do you think cyclic quadrilateral submerge to the theory that angle at centre of circle is twice the angle at the circumference of a circle.

The end