10.1 Tangents to Circles

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10.1 Tangents to Circles
Geometry
Objectives/Assignment
• Recall the basic properties of circle.
• Identify segments and lines related
to circles.
• Use properties of a tangent to a
circle.
• Assignment:
Worksheet and Problems from Exercise
not done in class.
Some definitions you need
• 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 “circle
P”, or
P.
• 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.
Some definitions you need
• The distance across
the circle, through its
center is the diameter
of the circle. The
diameter is twice the
radius.
• The terms radius and
diameter describe
segments as well as
measures.
center
diameter
radius
Some definitions you need
• A radius is a
segment whose
endpoints are the
center of the circle
and a point on the
circle.
• QP, QR, and QS
are radii of
Q.
All radii of a circle
are congruent.
S
P
Q
R
Some definitions you need
• A chord is a
segment whose
endpoints are
points on the
circle. PS and PR
are chords.
• A diameter is a
chord that passes
through the center
of the circle. PR is
a diameter.
S
P
Q
R
1.Secant of the circle
• A line which
intersects a circle
at two distinct
points is called a
secant of the
circle.
B
A
What is the difference
between a chord and a
secant of the circle ?
B
B
A
A
2.Tangent of the circle
• A line which
intersects(meets)
the circle at
exactly one point
is called a tangent
of the circle.
A
Ex. 1: Identifying Special
Segments and Lines
Tell whether the line or
segment is best
described as a chord, a
secant, a tangent, a
diameter, or a radius of
the circle.
a. AD
b. CD
c. EG
d. HB
K
B
A
J
C
D
E
H
F
G
3.Concentric circles
• Circles that
have a
common center
are called
concentric
circles.
No points of
intersection
Concentric
circles
Using properties of tangents
4.The point at which a tangent line
intersects(meets or touches) the
circle to which it is tangent is called
the point of contact or point of
tangency.
5.Thus, the point of contact is the
common point of the tangent to the
circle and the circle.
6.Theorem 10.1
• The tangent at any
point of a circle is
perpendicular to the
radius through the
point of contact.
• If l is tangent to the
circle with centre Q at
point P, then l ⊥QP.
P
Q
l
7.Conversely,
• In a plane, if a line is
perpendicular to a
radius of a circle at
its endpoint on a
circle, then the line
is tangent to the
circle.
• If l ⊥QP at P, then l is
tangent to the circle.
P
Q
l
8.Also,
• Since there can be only one
perpendicular to QP at the
point P, it follows that one
and only one tangent can
be drawn to a circle at a
given point on the
circumference.
• In fact to draw a tangent to
the circle at P, join QP and
draw a line perpendicular to
QP at P.
P
Q
l
9. And
• Since there can be only one
perpendicular to l at the
point P, it follows that the
perpendicular to a tangent
at its point of contact
passes through the centre.
P
Q
l
10. Prove that the tangents drawn at
the ends of a diameter of a circle are
parallel.
• Given:
• To prove:
• Proof:
l
11. Prove that the segment joining the
points of contact of two parallel
tangents of a circle, passes through
the centre.
• Given:
• To prove:
• Proof:
Ex. 4: Verifying a Tangent to a
Circle
• In the figure, D is the
centre of the circle. E
is a point on the
circle.
F is a point outside
the circle such that
DF = 61 units and EF
= 60 units.
Is EF a tangent to the
circle?
D
61
11
E
60
F
Ex. 5: Finding the radius of a
circle
• 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?
B
16 ft.
r
C
8 ft.
A
r
B
16 ft.
Solution:
r
A
c2 = a2 + b2
(r + 8)2 = r2 + 162
r 2 + 16r + 64 = r2 + 256
16r + 64 = 256
16r = 192
r = 12
C
8 ft.
r
Pythagorean Thm.
Substitute values
Square of binomial
Subtract r2 from each side.
Subtract 64 from each side.
Divide.
The radius of the silo is 12 feet.
Prove that in two concentric circles,
the chord of the larger circle, which touches
the
smaller circle, is bisected at the point of
contact
• Two concentric circles are of radii 5
cm and 3 cm. Find the length of the
chord of the larger circle which
touches the smaller circle.
NUMBER OF TANGENTS
TO A CIRCLE PASSING
THROUGH A POINT
(say P)
Case 1: P is inside the circle
The number of
tangents possible is
……………
Case 2: P is on the circle
• Recall point no. 8
• Since there can be
only one perpendicular
to QP at the point P, it
follows that one and
only one tangent can
be drawn to a circle at
a given point on the
circumference.
P
Q
Case 3. P is outside the
circle
• From a point in the circle’s exterior,
you can draw exactly two different
tangents to the circle.
• Recall : Angle in a semicircle is a
right angle.
• Or the angle subtended by a
diameter on the circumference is a
right angle.
Length of a tangent from a
point outside the circle
• It is the distance between the point
and the point of contact. In the figure
below PT is the length of the tangent.
T
P
Theorem 10.3:
The lengths of tangents drawn from an
external point to a circle are equal.
•
Chord of contact
• The line segment joining the points
of contact of two tangents drawn
from a point outside the circle is
called a chord of contact.

• In the figure above, prove the following:
1. OP bisects angle APB.
2. OAPB is a cyclic quadrilateral.
3. Angle APB and angle AOB are supplementary.
4. Angle APB = 2 (angle BAO).
5. OP bisects AB at right angles.
6. If AB = 8 cm and OA = 5 cm, find PA.
Circle inscribed in a triangle or
A triangle circumscribing a circle.
• Let r be the radius of
the circle and a, b, c
be the sides of the
triangle.
s
abc
2
• Show that area of
triangle = r X s.
• s= BD + CE + AF
• A triangle ABC is drawn to circumscribe a circle
of radius 4 cm such that the segments BD and
DC into which BC is divided by the point of
contact D are of lengths 8 cm and 6 cm
respectively. Find the sides AB and AC.
Circle inscribed in a quadrilateral or
A quadrilateral circumscribing a circle.
• Show that
AB + CD = BC + AD
• Parallelogram
subscribing a circle is a
rhombus.
• Opposite sides of a
quadrilateral
circumscribing a circle
subtend supplementary
angles at the centre of
the circle.
Textbook problem no. 9
• In Fig., XY and X′Y′ are
two parallel tangents to a
circle with centre O and
another tangent AB with
point of contact C
intersecting XY at A and
X′Y′ at B. Prove
that ∠ AOB = 90°.

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