This document discusses various concepts related to circles. It defines key terms like radius, diameter, circumference, chord, arc, secant, and tangent. It also presents two theorems - that the tangent at any point of a circle is perpendicular to the radius through that point, and that the lengths of two tangents drawn from an external point to a circle are equal. Examples of different types of intersections between circles and lines are provided. The document concludes by restating the key definitions and concepts covered.

1. By:- SAURAV RANJAN X

2.  About Circles….
 Its components….
 Theorem 10.1
 Questions
 Theorem 10.2
 Questions
2

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
Minor
Segment
Major
Segment
1
2

6. 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.

7. If the line segment that forms a chord of a circle is
extended on both the sides, the straight line with two
points on the circle is known as a secant.
O
A
B
Chord
Secant

8. Tangent to a circle is a line that intersects the circle
at only one point.
O
A
B
Tangent

9. Let examine the different
situations that can be
arise when a circle and a
line are given in a plane…
9

10. 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
A
B
M
N
O
A
B
P
Non-intersecting Line
Line AB and the
circle have no
common points

11. Here AB is the only tangent at point m of the circle….
O
A
B.m

12.  The tangent at any point of a circle is
perpendicular to the radius through the point of
contact.
12
90
Point of contact
Radius

13. 13
O
P Q
R
A B
We know that among all line segments joining
the point O to a point on AB, the shortest one
is I to AB. So , to prove the OP I AB , it is
sufficient to prove that OP is shorter then any
other segment joining O to any point of AB.

14. 14
O
P Q
R
A B
OP=OR (Radii of the same
circle)
Now, OQ=OR+RQ
→ OQ>OR
→ OQ>OP
Thus, OP is shorter than
any other segment joining
O to any point of AB.
Hence , OP I AB.

15. Theorem 10.2
 The lengths of two tangents drawn from an external
point to a circle are equal.
15
.
O
P
Q
A
O

16. Proof:-
In order to prove that AP=AQ , we shall first prove that ∆
OPA≈ ∆ OQA.
Since,OP I AP and OQ I AQ. (WHY)?
→ L OPA = L OQA=90 ____1
Now, in right triangles OPA and OQA , we have
OP=OQ (Radii of circle)
L OPA=L OQA (from __1)
OA=OA (Common)
∆ OPA≈ ∆ OQA (by RHS congruency)
→ AP = AQ
16

17. 17
.
O
P
Q
A
O

18. 18
O
P A
A
O
Q

19.  In a plane, two circles can
intersect in two points, one
point, or no points.
Coplanar circles that
intersect in one point are
called tangent circles.
Coplanar circles that have a
common center are called
concentric.
2 points of intersection.

20.  A line or segment that is
tangent to two coplanar
circles is called a common
tangent. A common internal
tangent intersects the
segment that joins the
centers of the two circles. A
common external tangent
does not intersect the
segment that joins the
center of the two circles.
Internally
tangent
Externally
tangent

21.  Circles that have
a common
center are called
concentric
circles.
Concentric
circles
No points of
intersection

22.  You are standing at C, 8 feet
away from a grain silo. The
distance from you to a point
of tangency is 16 feet. What
is the radius of the silo?
 First draw it. Tangent BC is
perpendicular to radius AB
at B, so ∆ABC is a right
triangle; so you can use
the Pythagorean theorem
to solve.
8 ft.
16 ft.
r
r
A
B
C

23. 8 ft.
16 ft.
r
r
A
B
C
(r + 8)2
= r2
+ 162
Pythagoras Thm.
Substitute values
c2
= a2
+ b2
r 2
+ 16r + 64 = r2
+ 256 Square of binomial
16r + 64 = 256
16r = 192
r = 12
Subtract r2
from each side.
Subtract 64 from each side.
Divide.
The radius of the silo is 12 feet.

24. Let us revise what we have learnt in this session.
 A circle is a collection of all points in a plane
which are at a constant distance from a fixed point.
 Radius is the distance from the center of a circle
to any point on the circle.
 Diameter is the distance across a circle through
the center.
 Circumference is the distance around the circle.
 A chord is a straight line segment with its end
points on the circumference of a circle

25.  An arc is a part of a circumference of a circle.
 If the line segment that forms a chord of a circle is
extended on both the sides, the straight line with two
points on the circle is known as a secant.
 Sector of a circle is a portion of a circle enclosed by
two radii and an arc.
 Intersecting line have two common points & Non-
intersecting line have no common points with the
circle.
 Tangent have only one common point with the circle.

26. THANK YOU
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