Textbook: Circle | Mathematics Class 10 (Maharashtra SSC Board) PDF Download

 Page 1


47
• Circles passing through one, two,three points • Secant and tangent
• Circles touching each other • Arc of a circle
• Inscribed angle and intercepted arc • Cyclic quadrilateral
• Secant tangent angle theorem • Theorem of intersecting chords
You are familiar with the concepts regarding circle, like - centre, radius, diameter, 
chord, interior and exterior of a circle. Also recall  the meanings of - congruent circles, 
concentric circles and intersecting circles. 
 Recall and write theorems and properties which are useful to find the solution 
of the above problem. 
(1)  The perpendicular drawn from centre to a chord 
(2) 
(3) 
 Using these properties, solve the above problem.
C
D E
F
Fig. 3.1
Activity I : In the adjoining figure, seg DE is  
  a chord of a circle with centre C. 
  seg CF ^ seg DE. If diameter of the 
  circle is 20 cm, DE =16 cm 
  find CF.
Recall the properties of chord studied in previous standard and perform the activity 
below.
congruent circles
concentric circles intersecting circles
3
Circle
Let’s study.
Let’s recall.
Page 2


47
• Circles passing through one, two,three points • Secant and tangent
• Circles touching each other • Arc of a circle
• Inscribed angle and intercepted arc • Cyclic quadrilateral
• Secant tangent angle theorem • Theorem of intersecting chords
You are familiar with the concepts regarding circle, like - centre, radius, diameter, 
chord, interior and exterior of a circle. Also recall  the meanings of - congruent circles, 
concentric circles and intersecting circles. 
 Recall and write theorems and properties which are useful to find the solution 
of the above problem. 
(1)  The perpendicular drawn from centre to a chord 
(2) 
(3) 
 Using these properties, solve the above problem.
C
D E
F
Fig. 3.1
Activity I : In the adjoining figure, seg DE is  
  a chord of a circle with centre C. 
  seg CF ^ seg DE. If diameter of the 
  circle is 20 cm, DE =16 cm 
  find CF.
Recall the properties of chord studied in previous standard and perform the activity 
below.
congruent circles
concentric circles intersecting circles
3
Circle
Let’s study.
Let’s recall.
48
Activity II : In the adjoining figure, 
 seg QR is a chord of the circle with  
 centre O. P is the midpoint of the  
 chord QR. If  QR = 24, OP = 10 , 
 find radius of the circle.
 To find solution of the problem, write the theorems that are useful.
(1)  
(2) 
 Using these theorems solve the problems.
Activity III :  In the adjoining figure, M  
  is the centre of the circle and  
  seg AB is a diameter.
  seg MS ^ chord AD
  seg MT ^ chord AC
  ÐDAB @ ÐCAB.
 Prove that : chord AD @ chord AC.
 To solve this problem which of the following theorems will you use ?
(1)  The chords which are equidistant from the centre are equal in length.
(2) Congruent chords of a circle are equidistant from the centre.
 Which of the following tests of congruence of triangles will be useful?
 (1) SAS, (2) ASA, (3) SSS, (4) AAS, (5) hypotenuse-side test.
 Using appropriate test and theorem write the proof of the above example.
Circles passing through one, two, three points
In the adjoining figure, point A lies in a plane. 
All the three circles with centres P, Q, R pass 
through point A. How many more such circles may 
pass through point A?
If your answer is many or innumerable, it is 
correct.
Infinite number of  circles pass through a point.
Fig. 3.2
P
Q R
O
A
T
D
B
C
S
M
Fig. 3.4
P
A
Q
R
Fig. 3.3
Let’s learn.
Page 3


47
• Circles passing through one, two,three points • Secant and tangent
• Circles touching each other • Arc of a circle
• Inscribed angle and intercepted arc • Cyclic quadrilateral
• Secant tangent angle theorem • Theorem of intersecting chords
You are familiar with the concepts regarding circle, like - centre, radius, diameter, 
chord, interior and exterior of a circle. Also recall  the meanings of - congruent circles, 
concentric circles and intersecting circles. 
 Recall and write theorems and properties which are useful to find the solution 
of the above problem. 
(1)  The perpendicular drawn from centre to a chord 
(2) 
(3) 
 Using these properties, solve the above problem.
C
D E
F
Fig. 3.1
Activity I : In the adjoining figure, seg DE is  
  a chord of a circle with centre C. 
  seg CF ^ seg DE. If diameter of the 
  circle is 20 cm, DE =16 cm 
  find CF.
Recall the properties of chord studied in previous standard and perform the activity 
below.
congruent circles
concentric circles intersecting circles
3
Circle
Let’s study.
Let’s recall.
48
Activity II : In the adjoining figure, 
 seg QR is a chord of the circle with  
 centre O. P is the midpoint of the  
 chord QR. If  QR = 24, OP = 10 , 
 find radius of the circle.
 To find solution of the problem, write the theorems that are useful.
(1)  
(2) 
 Using these theorems solve the problems.
Activity III :  In the adjoining figure, M  
  is the centre of the circle and  
  seg AB is a diameter.
  seg MS ^ chord AD
  seg MT ^ chord AC
  ÐDAB @ ÐCAB.
 Prove that : chord AD @ chord AC.
 To solve this problem which of the following theorems will you use ?
(1)  The chords which are equidistant from the centre are equal in length.
(2) Congruent chords of a circle are equidistant from the centre.
 Which of the following tests of congruence of triangles will be useful?
 (1) SAS, (2) ASA, (3) SSS, (4) AAS, (5) hypotenuse-side test.
 Using appropriate test and theorem write the proof of the above example.
Circles passing through one, two, three points
In the adjoining figure, point A lies in a plane. 
All the three circles with centres P, Q, R pass 
through point A. How many more such circles may 
pass through point A?
If your answer is many or innumerable, it is 
correct.
Infinite number of  circles pass through a point.
Fig. 3.2
P
Q R
O
A
T
D
B
C
S
M
Fig. 3.4
P
A
Q
R
Fig. 3.3
Let’s learn.
49
In the adjoining figure, how many circles 
pass through points A and B? 
How many circles contain all the three points 
A, B, C?
Perform the activity given below and try to 
find the answer.
ActivityI: Draw segment AB. Draw 
  perpendicular bisector l of the 
  segment AB. Take point P on  
  the line l as centre, PA as radius  
  and draw a circle. Observe that  
  the circle passes through point B 
   also. Find the reason. (Recall  
  the property of perpendicular  
  bisector of a segment.)
Taking any other point Q on the line l, if a circle is drawn with centre Q and radius 
QA, will it pass through B ? Think. 
How many such circles can be drawn, passing through A and B ? Where will their 
centres lie?
Activity II : Take any three non-collinear  
 points. What should be done to  
 draw a circle passing through all  
 these points ? Draw a circle  
 passing through these points. 
 Is it possible to draw one more  
 circle passing through these three  
 points ? Think of it.
Activity III : Take 3 collinear points D, E, F. Try to draw a circle passing through  
 these points. If you are not able to draw a circle, think of the reason.
(1) Infinite  circles pass through one point.
(2) Infinite circles pass through two distinct points.
(3) There is a unique circle passing through three non-collinear points.
(4) No circle can pass through 3 collinear points.
Fig. 3.5
Fig. 3.6
Fig. 3.7
A
B
C
A B
C
P
Q
A B
l
Let’s recall.
Page 4


47
• Circles passing through one, two,three points • Secant and tangent
• Circles touching each other • Arc of a circle
• Inscribed angle and intercepted arc • Cyclic quadrilateral
• Secant tangent angle theorem • Theorem of intersecting chords
You are familiar with the concepts regarding circle, like - centre, radius, diameter, 
chord, interior and exterior of a circle. Also recall  the meanings of - congruent circles, 
concentric circles and intersecting circles. 
 Recall and write theorems and properties which are useful to find the solution 
of the above problem. 
(1)  The perpendicular drawn from centre to a chord 
(2) 
(3) 
 Using these properties, solve the above problem.
C
D E
F
Fig. 3.1
Activity I : In the adjoining figure, seg DE is  
  a chord of a circle with centre C. 
  seg CF ^ seg DE. If diameter of the 
  circle is 20 cm, DE =16 cm 
  find CF.
Recall the properties of chord studied in previous standard and perform the activity 
below.
congruent circles
concentric circles intersecting circles
3
Circle
Let’s study.
Let’s recall.
48
Activity II : In the adjoining figure, 
 seg QR is a chord of the circle with  
 centre O. P is the midpoint of the  
 chord QR. If  QR = 24, OP = 10 , 
 find radius of the circle.
 To find solution of the problem, write the theorems that are useful.
(1)  
(2) 
 Using these theorems solve the problems.
Activity III :  In the adjoining figure, M  
  is the centre of the circle and  
  seg AB is a diameter.
  seg MS ^ chord AD
  seg MT ^ chord AC
  ÐDAB @ ÐCAB.
 Prove that : chord AD @ chord AC.
 To solve this problem which of the following theorems will you use ?
(1)  The chords which are equidistant from the centre are equal in length.
(2) Congruent chords of a circle are equidistant from the centre.
 Which of the following tests of congruence of triangles will be useful?
 (1) SAS, (2) ASA, (3) SSS, (4) AAS, (5) hypotenuse-side test.
 Using appropriate test and theorem write the proof of the above example.
Circles passing through one, two, three points
In the adjoining figure, point A lies in a plane. 
All the three circles with centres P, Q, R pass 
through point A. How many more such circles may 
pass through point A?
If your answer is many or innumerable, it is 
correct.
Infinite number of  circles pass through a point.
Fig. 3.2
P
Q R
O
A
T
D
B
C
S
M
Fig. 3.4
P
A
Q
R
Fig. 3.3
Let’s learn.
49
In the adjoining figure, how many circles 
pass through points A and B? 
How many circles contain all the three points 
A, B, C?
Perform the activity given below and try to 
find the answer.
ActivityI: Draw segment AB. Draw 
  perpendicular bisector l of the 
  segment AB. Take point P on  
  the line l as centre, PA as radius  
  and draw a circle. Observe that  
  the circle passes through point B 
   also. Find the reason. (Recall  
  the property of perpendicular  
  bisector of a segment.)
Taking any other point Q on the line l, if a circle is drawn with centre Q and radius 
QA, will it pass through B ? Think. 
How many such circles can be drawn, passing through A and B ? Where will their 
centres lie?
Activity II : Take any three non-collinear  
 points. What should be done to  
 draw a circle passing through all  
 these points ? Draw a circle  
 passing through these points. 
 Is it possible to draw one more  
 circle passing through these three  
 points ? Think of it.
Activity III : Take 3 collinear points D, E, F. Try to draw a circle passing through  
 these points. If you are not able to draw a circle, think of the reason.
(1) Infinite  circles pass through one point.
(2) Infinite circles pass through two distinct points.
(3) There is a unique circle passing through three non-collinear points.
(4) No circle can pass through 3 collinear points.
Fig. 3.5
Fig. 3.6
Fig. 3.7
A
B
C
A B
C
P
Q
A B
l
Let’s recall.
50
circle. At the end, they get merged in point P, but the angle between the radius OP and 
line AB will remain a right angle.  
At this stage the line AB becomes a tangent of the circle at P. 
So it is clear that, the tangent at any point of a circle is perpendicular to the radius 
at that point. 
This property is known as ‘tangent theorem’.
Two points Q and R are common to both, the line n and the circle with centre C. 
Q and R are intersecting points of line n and the circle. Line n is called a secant of  
the circle .
 Let us understand an important property of a tangent from the following activity.
Fig. 3.8
Fig. 3.9
A
R
B
l m
n
C
P
Q
A
P
O
B
In the  figure above, not a single point is common in line l  and circle with centre A. 
Point P is common to both, line m and circle with centre B. Here, line m is called a    
 tangent of the circle and point P is called the point of contact.
Let’s learn.
Secant and tangent
Activity :
Draw a sufficiently large circle with 
centre O. Draw radius OP. Draw a line 
AB ^ seg OP. It intersects the circle at 
points A, B. Imagine the line slides 
towards point P such that all the time it 
remains parallel to its original position. 
Obviously, while the line slides, points 
A and B approach each other along the 
Page 5


47
• Circles passing through one, two,three points • Secant and tangent
• Circles touching each other • Arc of a circle
• Inscribed angle and intercepted arc • Cyclic quadrilateral
• Secant tangent angle theorem • Theorem of intersecting chords
You are familiar with the concepts regarding circle, like - centre, radius, diameter, 
chord, interior and exterior of a circle. Also recall  the meanings of - congruent circles, 
concentric circles and intersecting circles. 
 Recall and write theorems and properties which are useful to find the solution 
of the above problem. 
(1)  The perpendicular drawn from centre to a chord 
(2) 
(3) 
 Using these properties, solve the above problem.
C
D E
F
Fig. 3.1
Activity I : In the adjoining figure, seg DE is  
  a chord of a circle with centre C. 
  seg CF ^ seg DE. If diameter of the 
  circle is 20 cm, DE =16 cm 
  find CF.
Recall the properties of chord studied in previous standard and perform the activity 
below.
congruent circles
concentric circles intersecting circles
3
Circle
Let’s study.
Let’s recall.
48
Activity II : In the adjoining figure, 
 seg QR is a chord of the circle with  
 centre O. P is the midpoint of the  
 chord QR. If  QR = 24, OP = 10 , 
 find radius of the circle.
 To find solution of the problem, write the theorems that are useful.
(1)  
(2) 
 Using these theorems solve the problems.
Activity III :  In the adjoining figure, M  
  is the centre of the circle and  
  seg AB is a diameter.
  seg MS ^ chord AD
  seg MT ^ chord AC
  ÐDAB @ ÐCAB.
 Prove that : chord AD @ chord AC.
 To solve this problem which of the following theorems will you use ?
(1)  The chords which are equidistant from the centre are equal in length.
(2) Congruent chords of a circle are equidistant from the centre.
 Which of the following tests of congruence of triangles will be useful?
 (1) SAS, (2) ASA, (3) SSS, (4) AAS, (5) hypotenuse-side test.
 Using appropriate test and theorem write the proof of the above example.
Circles passing through one, two, three points
In the adjoining figure, point A lies in a plane. 
All the three circles with centres P, Q, R pass 
through point A. How many more such circles may 
pass through point A?
If your answer is many or innumerable, it is 
correct.
Infinite number of  circles pass through a point.
Fig. 3.2
P
Q R
O
A
T
D
B
C
S
M
Fig. 3.4
P
A
Q
R
Fig. 3.3
Let’s learn.
49
In the adjoining figure, how many circles 
pass through points A and B? 
How many circles contain all the three points 
A, B, C?
Perform the activity given below and try to 
find the answer.
ActivityI: Draw segment AB. Draw 
  perpendicular bisector l of the 
  segment AB. Take point P on  
  the line l as centre, PA as radius  
  and draw a circle. Observe that  
  the circle passes through point B 
   also. Find the reason. (Recall  
  the property of perpendicular  
  bisector of a segment.)
Taking any other point Q on the line l, if a circle is drawn with centre Q and radius 
QA, will it pass through B ? Think. 
How many such circles can be drawn, passing through A and B ? Where will their 
centres lie?
Activity II : Take any three non-collinear  
 points. What should be done to  
 draw a circle passing through all  
 these points ? Draw a circle  
 passing through these points. 
 Is it possible to draw one more  
 circle passing through these three  
 points ? Think of it.
Activity III : Take 3 collinear points D, E, F. Try to draw a circle passing through  
 these points. If you are not able to draw a circle, think of the reason.
(1) Infinite  circles pass through one point.
(2) Infinite circles pass through two distinct points.
(3) There is a unique circle passing through three non-collinear points.
(4) No circle can pass through 3 collinear points.
Fig. 3.5
Fig. 3.6
Fig. 3.7
A
B
C
A B
C
P
Q
A B
l
Let’s recall.
50
circle. At the end, they get merged in point P, but the angle between the radius OP and 
line AB will remain a right angle.  
At this stage the line AB becomes a tangent of the circle at P. 
So it is clear that, the tangent at any point of a circle is perpendicular to the radius 
at that point. 
This property is known as ‘tangent theorem’.
Two points Q and R are common to both, the line n and the circle with centre C. 
Q and R are intersecting points of line n and the circle. Line n is called a secant of  
the circle .
 Let us understand an important property of a tangent from the following activity.
Fig. 3.8
Fig. 3.9
A
R
B
l m
n
C
P
Q
A
P
O
B
In the  figure above, not a single point is common in line l  and circle with centre A. 
Point P is common to both, line m and circle with centre B. Here, line m is called a    
 tangent of the circle and point P is called the point of contact.
Let’s learn.
Secant and tangent
Activity :
Draw a sufficiently large circle with 
centre O. Draw radius OP. Draw a line 
AB ^ seg OP. It intersects the circle at 
points A, B. Imagine the line slides 
towards point P such that all the time it 
remains parallel to its original position. 
Obviously, while the line slides, points 
A and B approach each other along the 
51
Tangent theorem
Theorem : A tangent at any point of a circle is perpendicular to the radius at  
    the point of contact. 
    There is an indirect proof of this theorem.
For more information
Given : Line l is a tangent to the circle with centre O at the point of contact A.
To prove : line l ^ radius OA.
Proof : Assume that, line l is not  
  perpendicular  to seg OA.
  Suppose, seg OB is drawn  
  perpendicular to line l. 
  Of course B is not same as A.  
  Now take a point C on line l 
  such that A-B-C and 
  BA = BC .
  Now in, D OBC and D OBA
  seg BC @ seg BA ........ (construction)
  Ð OBC @ Ð OBA ....... (each right angle)
  seg OB @ seg OB
 \ D OBC @ D OBA .......... (SAS test)
 \ OC = OA
  But seg OA is a radius.
 \seg OC must also be radius.
 \ C lies on the circle.
  That means line l intersects the circle in two  
  distinct points A and C.
  But line l is a tangent. ........... (given)
 \ it intersects the circle in only one point.
O A
l
Fig. 3.10
Fig. 3.11
A
B
C
O
l
  Our assumption that line l is not perpendicular to radius OA is wrong.
 \ line l ^ radius OA.