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Introduction to geometry

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Christopher Chibangu
Christopher Chibangu

The document discusses fractions of revolutions and measuring angles. It explains that a full revolution is divided into 360 degrees. When the second hand of a clock moves from one hour to another, it turns through a fraction of a revolution or number of degrees. Examples show calculating fractions of revolutions and measuring angles in degrees using a protractor. Different types of angles such as acute, obtuse, right and reflex angles are also defined.

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Introduction to Geometry 6/14/2018 zmaths4all
Fraction of a revolution When the second hand of a clock starts at 12 and moves round until it stops at 12 again, it has gone through one complete turn. One complete turn is called a revolution. When the seconds hand starts at 12 and stops at 3, it has turned through 1 4 of a revolution. 6/14/2018 zmaths4all
Worked Example 10a What fraction of a revolution does the seconds hand of a clock turn through when It starts at 3 and stops at 12 It starts at 4 and stops at 8 If the seconds hand starts at 3 and stops at 12, it turns through 3 4 of a revolution. If the seconds hand starts at 4 and stops at 8, it turns through 1 3 of a revolution. 6/14/2018 zmaths4all
Questions What fraction of a revolution does the second hand of a clock turn through when it (show by drawing second hand in the circle) a) Starts at 12 and stops at 6 Ans: Half ( 1 2 ) of a revolution b) Start at 7 and stops at 10 Ans: 1 4 of a revolution c) Start at 5 and stops at 2 Ans: 3 4 of a revolution 6/14/2018 zmaths4all
Worked example Where does the hand stop if it starts at 12 and turns through 1 4 of a revolution Ans: It stops at 3 A revolution has 12 steps 1 4 of 12 is 3. 3 steps from 12 is 3. 6/14/2018 zmaths4all
Questions Where does the hand stop if a) It starts at 12 and turns through 3 4 of a complete turn Ans: It stops at 9 b) It starts at 9 and turns trough a 1 2 a complete turn Ans: It stops at 3 c) It starts at 12 and turns through 2 3 of a revolution Ans: It stops at 8 6/14/2018 zmaths4all
Compass directions The four main compass directions are North, South, East and West. Question If you are facing West and turn through 1 2 3 revolutions, in Which direction are you facing? 6/14/2018 zmaths4all If you stand facing North, and turn clockwise through 1 2 a revolution, you are then facing South
Angles ? When the hand of a clock moves from one position to another it has turned trough an angle ? A quarter of a revolution is called a right angle ? Half of a revolution is two right angle 6/14/2018 zmaths4all Right angle Two right angles
Worked example How many right angles does the second hand of a clock turn through when it starts at 3 and stops at 12 6/14/2018 zmaths4all Ans: It turns trough three right Questions How many right angle does the seconds hand of a clock turn through when a) Starts at 6 and stops at 9 b) Start at 8 and stops at 2 c) How many right angles do you turn trough if you face North and turn anticlockwise to face East?
Measuring angles One complete revolution is divided into 360 parts. Each part is called a degree. 360 degrees is 3600 6/14/2018 zmaths4all Worked example How many degrees has the seconds hand of a clock turned through when it moves from 6 to 8? When the hand moves from 6 to 8 it turns through 2 12 revolution Ans; The hand turns through 2 12 of 3600 which is 600 8 6
Using protractors to measure angles A protractor looks like this 6/14/2018 zmaths4all It has the following - Base line - Center - scale From the base line, you read anticlockwise on the scale from 00 to 700 . The angle is 700 Measure this angle
Questions Measure the following angles; 6/14/2018 zmaths4all ?? ? ??? ?
Worked example a) Find the size of the angle marked p 6/14/2018 zmaths4all Angle p and 400 together make 3600 So angle p is 3600- 400 = 3200 400 P b) Measure the following angle This is a reflex angle and it is bigger than 3 right angles, i.e it is greater than 2700 To find this angle, we need to measure the smaller angle, marked p Angle p is 550 , so the reflex angle is 3600 - 550 = 3050 P
Drawing angles using a protractor To draw an angle of 1200, start by drawing one arm and mark the corner of your angle. The corner, or point, of an angle is called vertex. 6/14/2018 zmaths4all Place the protractor as shown in the diagram. Make sure that the vertex is at the center of the base line. Choose the scale that starts at 00 on the drawn line, and mark the paper next to the 1200 mark on the scale. Remove the protractor and join your mark to the vertex.
Types of Angles Any angle that is smaller than a right angle is called acute angle 6/14/2018 zmaths4all Any angle that is greater than one right angle and less than two right angles is called an obtuse angle Any angle that is greater than two right angles is called a reflex angle. A right is 900
Related angles Vertically opposite angles When two straight lines cross, four angles are formed. 6/14/2018 zmaths4all The two angles that are opposite each other are called vertically opposite angles . Vertically opposite angles are equal P r qs

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Introduction to geometry

  • 2. Fraction of a revolution When the second hand of a clock starts at 12 and moves round until it stops at 12 again, it has gone through one complete turn. One complete turn is called a revolution. When the seconds hand starts at 12 and stops at 3, it has turned through 1 4 of a revolution. 6/14/2018 zmaths4all
  • 3. Worked Example 10a What fraction of a revolution does the seconds hand of a clock turn through when It starts at 3 and stops at 12 It starts at 4 and stops at 8 If the seconds hand starts at 3 and stops at 12, it turns through 3 4 of a revolution. If the seconds hand starts at 4 and stops at 8, it turns through 1 3 of a revolution. 6/14/2018 zmaths4all
  • 4. Questions What fraction of a revolution does the second hand of a clock turn through when it (show by drawing second hand in the circle) a) Starts at 12 and stops at 6 Ans: Half ( 1 2 ) of a revolution b) Start at 7 and stops at 10 Ans: 1 4 of a revolution c) Start at 5 and stops at 2 Ans: 3 4 of a revolution 6/14/2018 zmaths4all
  • 5. Worked example Where does the hand stop if it starts at 12 and turns through 1 4 of a revolution Ans: It stops at 3 A revolution has 12 steps 1 4 of 12 is 3. 3 steps from 12 is 3. 6/14/2018 zmaths4all
  • 6. Questions Where does the hand stop if a) It starts at 12 and turns through 3 4 of a complete turn Ans: It stops at 9 b) It starts at 9 and turns trough a 1 2 a complete turn Ans: It stops at 3 c) It starts at 12 and turns through 2 3 of a revolution Ans: It stops at 8 6/14/2018 zmaths4all
  • 7. Compass directions The four main compass directions are North, South, East and West. Question If you are facing West and turn through 1 2 3 revolutions, in Which direction are you facing? 6/14/2018 zmaths4all If you stand facing North, and turn clockwise through 1 2 a revolution, you are then facing South
  • 8. Angles ? When the hand of a clock moves from one position to another it has turned trough an angle ? A quarter of a revolution is called a right angle ? Half of a revolution is two right angle 6/14/2018 zmaths4all Right angle Two right angles
  • 9. Worked example How many right angles does the second hand of a clock turn through when it starts at 3 and stops at 12 6/14/2018 zmaths4all Ans: It turns trough three right Questions How many right angle does the seconds hand of a clock turn through when a) Starts at 6 and stops at 9 b) Start at 8 and stops at 2 c) How many right angles do you turn trough if you face North and turn anticlockwise to face East?
  • 10. Measuring angles One complete revolution is divided into 360 parts. Each part is called a degree. 360 degrees is 3600 6/14/2018 zmaths4all Worked example How many degrees has the seconds hand of a clock turned through when it moves from 6 to 8? When the hand moves from 6 to 8 it turns through 2 12 revolution Ans; The hand turns through 2 12 of 3600 which is 600 8 6
  • 11. Using protractors to measure angles A protractor looks like this 6/14/2018 zmaths4all It has the following - Base line - Center - scale From the base line, you read anticlockwise on the scale from 00 to 700 . The angle is 700 Measure this angle
  • 12. Questions Measure the following angles; 6/14/2018 zmaths4all ?? ? ??? ?
  • 13. Worked example a) Find the size of the angle marked p 6/14/2018 zmaths4all Angle p and 400 together make 3600 So angle p is 3600- 400 = 3200 400 P b) Measure the following angle This is a reflex angle and it is bigger than 3 right angles, i.e it is greater than 2700 To find this angle, we need to measure the smaller angle, marked p Angle p is 550 , so the reflex angle is 3600 - 550 = 3050 P
  • 14. Drawing angles using a protractor To draw an angle of 1200, start by drawing one arm and mark the corner of your angle. The corner, or point, of an angle is called vertex. 6/14/2018 zmaths4all Place the protractor as shown in the diagram. Make sure that the vertex is at the center of the base line. Choose the scale that starts at 00 on the drawn line, and mark the paper next to the 1200 mark on the scale. Remove the protractor and join your mark to the vertex.
  • 15. Types of Angles Any angle that is smaller than a right angle is called acute angle 6/14/2018 zmaths4all Any angle that is greater than one right angle and less than two right angles is called an obtuse angle Any angle that is greater than two right angles is called a reflex angle. A right is 900
  • 16. Related angles Vertically opposite angles When two straight lines cross, four angles are formed. 6/14/2018 zmaths4all The two angles that are opposite each other are called vertically opposite angles . Vertically opposite angles are equal P r qs