Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
SlideShare a Scribd company logo

Circles

β€’Download as PPTX, PDFβ€’
β€’
Varun Devang
Varun Devang

- A circle is a shape where all points are equidistant from the center point. Key terms include radius, diameter, chord, arc, sector, circumference, and tangent. - The circumference of a circle is the distance around it and is calculated as 2Ο€r. The area is calculated as Ο€r^2. - A tangent line to a circle is perpendicular to the radius drawn at the point of tangency. The lengths of tangents drawn from an external point to a circle are equal. - Circles are abundant in nature and everyday life, appearing in architecture, vehicles, wheels, balls, and many sports. Without circles, common motions and activities would not be possible.

1 of 17
B y: Va ru n
β€’ Circle – set of all points in a plane that are equidistant from a given point called a center of the circle. A circle with center P is called β€œcircle P”, or P. β€’ The distance from the center to a point on the circle is called the radius of the circle. Two circles are congruent if they have the same radius.
O = = = A C D EB Radius Diameter Centre of circle Circumference A circle is a shape with all points at the same distance from its centre. It’s the distance across a circle through the center. It’s the distance from the center of a circle to any point on the circle. All points on the circle are at same distance from the centre point. It’s the distance around the circle.
O A EB Chord Chord Diameter, the biggest chord Interior of the circle Exterior of the circle A chord is a straight line segment with its end points on the circumference of a circle
O M N Chord Major Arc An arc is a part of a circumference of a circle. Minor Arc 1 2
A chord divides the circular region into 2 parts, each of which is called a segment of the circle. D E Chord DE Minor segment of a circle Major segment of a circle S R οƒ˜ The part of the circular region enclosed by a major arc and the chord is called a major segment. οƒ˜ Minor segment does not contain the centre of the circle.
O A B Radius Radius Minor Sector Major Sector Sector of a circle is a portion of a circle enclosed by two radii and an arc.
O Secant Line AB and the circle have two common points M and N Tangent There is only one point P which is common to the line AB and the circle A B O M N O P Non-intersecting Line Line AB and the circle have no common points
Circles
The circumference of a circle is the distance around it. The term is used when measuring physical objects, as well as when considering abstract geometric forms. The area of a circle is the number of square units inside that circle. A = pr C = 2pr 2
Q P β€’ IF A LINE IS TANGENT TO A CIRCLE, THEN IT IS PERPENDICULAR TO THE RADIUS DRAWN TO THE POINT OF TANGENCY. β€’ IF L IS TANGENT TO Q AT POINT P, THEN L βŠ₯QP. β€’ THE TANGENT AT ANY POINT OF A CIRCLE IS PERPENDICULAR TO THE RADIUS THROUGH THE POINT OF l
Construction: Draw a circle with centre O. Draw a tangent XY which touches point P at the circle. We need to prove that OP is perpendicular to XY. Take a point Q on XY other than P and join OQ. The point Q must lie outside the circle. Therefore, OQ is longer than the radius OP of the circle. That is, OQ > OP. Since this happens for every point on the line XY except the point P, OP is the shortest of all the distances of the point O to the points of XY. So OP is perpendicular to XY. Remarks: 1. By the theorem above, we can also conclude that at any point on a circle there can be one and only one tangent 2. The line containing the radius through the point of contact is also sometimes called β€˜normal’ to the point Proof:
The lengths of tangents drawn from an external point to a circle are equal. Proof : We are given a circle with centre O, a point P lying outside the circle and two tangents PQ, PR on the circle from P. We are required to prove that PQ = PR. For this, we join OP, OQ and OR. Then angle OQP and angle ORP are right angles, because these are angles between the radii and tangents, and according to Theorem 10.1 they are right angles. Now in right triangles OQP and ORP, OQ = OR (Radii of the same circle) OP = OP (Common) Therefore, OQP ORP (RHS) This gives PQ = PR (CPCT)angle
Remarks : 1.The theorem can also be proved by using the Pythagoras Theorem as follows: 2PQ = 2OP – 2OQ = 2OP –2OR =2PR (As OQ = OR) which gives PQ = PR. Hence Proved 2. Note also that ∠ OPQ = ∠ OPR. Therefore, OP is the angle bisector of ∠ QPR, i.e., the centre lies on the bisector of the angle between the two tangents.
The circle is one of the most common shapes in our daily life, and indeed the universe. Planets, the movement of the planets, natural cycles, natural shapes - there are circles absolutely everywhere. The circle is one of the most complex shapes, and indeed the most difficult for man to create, yet nature manages to do it perfectly. The centres of flowers, eyes, and many more things are circular and we see them in our every day life.
However, circles are abundant in our daily life in regards to manmade objects, too. Circles can be found in modern architecture, in the house and out on the street. Without circles we wouldn't have cars. Cars only work because of the circular motion that is produced to make the circular wheels go round. Without circles the motion wouldn't be created and the wheels simply wouldn't move properly, or make the car move. If you're into sports, then a sudden lack of circles would completely ruin your sporting activities. In many sports, balls are necessary (football, tennis, ping pong, golf, and so many more). Without circles there would be no spheres, and these sports simply wouldn't be possible for anybody to play.
Circles

More Related Content

Circles

  • 2. β€’ Circle – set of all points in a plane that are equidistant from a given point called a center of the circle. A circle with center P is called β€œcircle P”, or P. β€’ The distance from the center to a point on the circle is called the radius of the circle. Two circles are congruent if they have the same radius.
  • 3. O = = = A C D EB Radius Diameter Centre of circle Circumference A circle is a shape with all points at the same distance from its centre. It’s the distance across a circle through the center. It’s the distance from the center of a circle to any point on the circle. All points on the circle are at same distance from the centre point. It’s the distance around the circle.
  • 4. O A EB Chord Chord Diameter, the biggest chord Interior of the circle Exterior of the circle A chord is a straight line segment with its end points on the circumference of a circle
  • 5. O M N Chord Major Arc An arc is a part of a circumference of a circle. Minor Arc 1 2
  • 6. A chord divides the circular region into 2 parts, each of which is called a segment of the circle. D E Chord DE Minor segment of a circle Major segment of a circle S R οƒ˜ The part of the circular region enclosed by a major arc and the chord is called a major segment. οƒ˜ Minor segment does not contain the centre of the circle.
  • 7. O A B Radius Radius Minor Sector Major Sector Sector of a circle is a portion of a circle enclosed by two radii and an arc.
  • 8. O Secant Line AB and the circle have two common points M and N Tangent There is only one point P which is common to the line AB and the circle A B O M N O P Non-intersecting Line Line AB and the circle have no common points
  • 10. The circumference of a circle is the distance around it. The term is used when measuring physical objects, as well as when considering abstract geometric forms. The area of a circle is the number of square units inside that circle. A = pr C = 2pr 2
  • 11. Q P β€’ IF A LINE IS TANGENT TO A CIRCLE, THEN IT IS PERPENDICULAR TO THE RADIUS DRAWN TO THE POINT OF TANGENCY. β€’ IF L IS TANGENT TO Q AT POINT P, THEN L βŠ₯QP. β€’ THE TANGENT AT ANY POINT OF A CIRCLE IS PERPENDICULAR TO THE RADIUS THROUGH THE POINT OF l
  • 12. Construction: Draw a circle with centre O. Draw a tangent XY which touches point P at the circle. We need to prove that OP is perpendicular to XY. Take a point Q on XY other than P and join OQ. The point Q must lie outside the circle. Therefore, OQ is longer than the radius OP of the circle. That is, OQ > OP. Since this happens for every point on the line XY except the point P, OP is the shortest of all the distances of the point O to the points of XY. So OP is perpendicular to XY. Remarks: 1. By the theorem above, we can also conclude that at any point on a circle there can be one and only one tangent 2. The line containing the radius through the point of contact is also sometimes called β€˜normal’ to the point Proof:
  • 13. The lengths of tangents drawn from an external point to a circle are equal. Proof : We are given a circle with centre O, a point P lying outside the circle and two tangents PQ, PR on the circle from P. We are required to prove that PQ = PR. For this, we join OP, OQ and OR. Then angle OQP and angle ORP are right angles, because these are angles between the radii and tangents, and according to Theorem 10.1 they are right angles. Now in right triangles OQP and ORP, OQ = OR (Radii of the same circle) OP = OP (Common) Therefore, OQP ORP (RHS) This gives PQ = PR (CPCT)angle
  • 14. Remarks : 1.The theorem can also be proved by using the Pythagoras Theorem as follows: 2PQ = 2OP – 2OQ = 2OP –2OR =2PR (As OQ = OR) which gives PQ = PR. Hence Proved 2. Note also that ∠ OPQ = ∠ OPR. Therefore, OP is the angle bisector of ∠ QPR, i.e., the centre lies on the bisector of the angle between the two tangents.
  • 15. The circle is one of the most common shapes in our daily life, and indeed the universe. Planets, the movement of the planets, natural cycles, natural shapes - there are circles absolutely everywhere. The circle is one of the most complex shapes, and indeed the most difficult for man to create, yet nature manages to do it perfectly. The centres of flowers, eyes, and many more things are circular and we see them in our every day life.
  • 16. However, circles are abundant in our daily life in regards to manmade objects, too. Circles can be found in modern architecture, in the house and out on the street. Without circles we wouldn't have cars. Cars only work because of the circular motion that is produced to make the circular wheels go round. Without circles the motion wouldn't be created and the wheels simply wouldn't move properly, or make the car move. If you're into sports, then a sudden lack of circles would completely ruin your sporting activities. In many sports, balls are necessary (football, tennis, ping pong, golf, and so many more). Without circles there would be no spheres, and these sports simply wouldn't be possible for anybody to play.

Editor's Notes

  1. Let us revise the concept of a circle. This is a point O in a plane. A circle is a collection of all points in a plane which are at a constant distance from the point O. If A, B, C, D and E are the points of the circle, then, length OA is equal to length OB is equal to length OC is equal to length OD is equal to length OE. Point O is called centre of the circle. OA, OB, OC, OD and OE are the radii of the circle. BE is the diameter of the circle and this is arc is the circumference of the circle.
  2. Audio Script: A chord is a straight line segment with its end points on the circumference of a circle. If we join any two points on the circle, it is called as chord of a circle. Here, AB, AE and BE are the Chords of the circle. Diameter is the biggest chord of a circle. The portion inside the circle is known as interior of the circle while the portion outside the circle is known as exterior of the circle.
  3. Audio Script: An arc is a part of a circumference of a circle. If segment MN is a chord, then, Arc MN-1 is known as the minor arc and arc MN-2 is known as the major arc. The interior region of a circle between a chord and an arc of a circle is known as a segment.
  4. Audio Script: Sector of a circle is a portion of a circle enclosed by two radii and an arc. The area enclosed by radii and major arc is known as Major Sector and the area enclosed by radii and minor arc is known as Minor Sector.
  5. Audio Script: Let us now observe the different situations that can arise when a circle and a line are given in a plane. Consider a circle with centre O and a line AB. There can be three possibilities. The line AB and the circle have no common point. Then, the line AB is called a non-intersecting line with respect to the circle. There are two common points M and N that the line AB and the circle have. That is, the line AB intersects the circle in two points, M and N. Then, the line AB is called as a secant of the circle. There is only one point P which is common to the line AB and the circle. Then, the line AB is called a tangent to the circle.