circle circular circumscribe circumference radius
radii
diameter intersect tangent secant vertex vertices
arc chord area quadrant approximately
2. Do you know the meaning of all the words? Discuss in pairs.
Find the words you do not know in the text on the next page and try to guess their meaning.
3. Discuss in pairs or groups.
• What is the difference between an ellipse and a circle?
• Give the definition of a circle.
• What is the difference between the circle and a disk?
• How can we find the area of a circle?
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THE CIRCLE
A circle* is a plane figure bounded by the curved line all points of which are the same distance away from the fixed point, called the centre or the point of origin. The length of the curved line, or the distance around the circle, is its circumference. A straight line drawn from the centre to any point on the circumference is called a radius (plural: radii). The line drawn from one side of the circle to the other, passing through the point of origin, is called the diameter.
Any part of the curved line which makes the circle (circumference) is an arc. A straight line joining the ends of an arc is called a chord. It connects any two points on the circumference of a circle. The part of a circle enclosed by an arc and a chord is called a segment. A part of a circle enclosed by two radii and an arc is called a sector. A line meeting the circumference but which (when produced) does not intersect it is called a tangent. A line which intersects the circumference in two places is called a secant.
The circles having the same point of origin are called concentric. A circle which passes through the vertices of a triangle is called the circumcircle of the triangle. And its center is called its circumcentre. The circle may be circumscribed around the triangle.
A half circle is called a semi-circle. When a circle is cut into four equal parts, each of the quarters is called a quadrant.
Area of a Circle. If a circle is divided into 4 quadrants and a square is circumscribed about the circle, four small squares are formed (Fig. 16.1).
Fig. 16.1
The side of each small square is equal to the radius r. The area of each small square is r2, the area of the large square is equal to the sum of the four small squares, or 4r2.
We can see that the area of the circle is about 4
3 of the large square. It is
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7 7
RULE to remember: the area of a circle equals π times the radius squared.
A= πr2.
NOTES TO THE TEXT
In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure (also known as the perimeter) or to the whole figure including its interior.
However, in strict technical usage, "circle"refers to the perimeter while the interior of the circle is called a disk.
Vocabulary and Language Focus
1. Translate into English . Use some of the expressions in sentences of your own.
1. kõverjoonega piiratud olema, 2. kindlaksmääratud punkt, 3. ringi keskpunkt, 4. ringi pindala mõõtma, 5. kaare ja kõõluga ümbritsetud olema, 6. ringjoont puutuv sirge, 7.
lõikuma, 8. puutuja ja lõikaja, 9. kolmnurga ümberringjoon, 10. raadius, raadiused, 11.
diameetri pikkus, 12. kaht punkti ühendama, 13. ligikaudselt võrdne olema, 14. kontsentriliste ringide keskpunkt, 15. neljaks veerandiks jagama, 16. kaare ja kahe raadiuse poolt ümbritsetud olema, 17. poolringi moodustama, 18. millestki võrdsel kaugusel olema, 19.
keskpunkti läbima, 20. nelja väikese ruudu summa.
2. Say whether the following statements are true (T) or false (F).
Correct the false statements.
a) A chord is a curved line.
b) The radius of a circle is half the length of its diameter.
c) A closed curve where all points on the curve are equidistant from the centre is called a circle.
d) A sector has three sides - two chords and arc.
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has been done for you.
Singular Plural
Vertex vertices
Index
Axis axes
Analysis Hypothesis Parenthesis Basis
Criterion criteria
Phenomenon
Formula formulae
Radius radii
4. Fill in the blanks.
1. All the __________ of a circle are equal. 2. __________ circles are circles which have the same __________ of __________. 3. A triangle has been __________ if a circle passes through its _________. 4. A __________ is the area enclosed by an arc and two __________ , while a __________ is the area __________ enclosed by an arc and a __________ . 5. If a line passes through a circle and intersects the circumference, it is called a __________, but a __________ meets the circumference without intersecting it. 6. A circle is a __________ geometric figure. 7. All the points on the circumference of a circle are __________ from the centre. 8. All the radii of a circle are __________. 9. The __________
bisector of a __________ passes through the centre of a circle. 10. We may think of a circle as a regular polygon __________ by an infinite number of sides. 11. The side of a circle is called the __________. 12. If we draw the __________ of a circle, the line divides the circle into two equal __________. 13. A __________ separates a circle into two __________ and the circumference into two __________. 14. A chord which passes through the centre of a circle is called a __________.
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1. Work in pairs. Name the following parts of the figure.
a) shaded area a b) shaded area b
c) EF d) XF e) AB
f) OC and OD g) O
2. Now answer these questions.
1. What are the properties of any triangle ABX with AB as its base and X as a point on the circumference?
2. When is a sector also a segment?
3. Give answers to the following tasks and questions.
1. A circle has a radius of 3 centimetres. Calculate a) the diameter
b) the circumference.
2. The circumference of a circle is approximately 15.7 centimetres. Calculate a) the approximate radius
b) the approximate diameter.
3. If two chords of a circle, AB and CD, intersect at 0, what is the relationship between AO · OB and CO · OD?
4. Ask and answer the following questions.
1. What is the area of a circle equal to? 2. What is the length of the bounding line of a circle called? 3. What is a radius? 4. What is a diameter? 5. What do we call a half of a circle?
6. What is the difference between a segment and a sector? 7. Can you explain the difference
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5. Work in pairs. Use the words you have learned to give information about these figures:
Group Work
Solve the puzzle.
The shape shown in the sketch below, obviously, is that of a square attached to half of another similar square, divided diagonally:
Can you divide it into four pieces all of precisely the same size and shape?
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