PinkMonkey.com Geometry Study Guide - CHAPTER 7 : CIRCLE
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CHAPTER 7 : CIRCLE
7.1 Introduction
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A circle is defined as a set of all such points in a given plane which lie at a fixed distance
from a fixed point in the plane. This fixed point is called the center of the circle and the
fixed distance is called the radius of the circle (see figure 7.1).
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Chapter 8
Figure 7.1
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Figure 8.1 shows a circle where point P is the center of the circle and segment PQ is known
as the radius. The radius is the distance between all points on the circle and P. It follows that
if a point R exists such that l (seg.PQ) > l (seg.PR) the R is inside the circle. On the other
hand for a point T if l (seg.PT) > l (seg.PQ) T lies outside the circle. In figure 8.1 since l (seg.
PS) = l (seg.PQ) it can be said that point S lies on the circle.
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7.1
Introduction
7.2 Lines of circle
7.3 Arcs
7.4 Inscribed
angels
7.5 Some
properties od
tangents,
secants and
chords
7.6 Chords and
their arcs
7.7 Segments of
chords secants
and tangents
7.8 Lengths of
arcs and area of
sectors
PinkMonkey.com Geometry Study Guide - 7.2 Lines of a Circle
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7.2 Lines of a Circle
The lines in the plane of the circle are classified into three categories (figure 7.2).
a) Lines like l which do not intersect the circle.
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b) Lines like m which intersect the circle at only one point.
c) Lines like n which intersect the circle at two points..
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7.1 Introduction
7.2 Lines of
circle
7.3 Arcs
7.4 Inscribed
angels
7.5 Some
properties od
tangents,
secants and
chords
7.6 Chords and
their arcs
7.7 Segments of
chords secants
and tangents
7.8 Lengths of
arcs and area of
sectors
Chapter 8
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Figure 8.2
Lines like m are called tangents. A tangent is a line that has one of its points on a circle and
the rest outside the circle.
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PinkMonkey.com Geometry Study Guide - 7.2 Lines of a Circle
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Line n is called a secant of the circle. A secant is defined as any line that intersects a circle
in two distinct points.
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A segment whose end points lie on a circle is called a Chord . In figure 7.2 AB is a chord of
the circle. Thus a chord is always a part of secant. A circle can have an infinite number of
chords of different lengths (figure 7.3)
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Figure 7.3
The longest chord of the circle passes through its center and is called as the diameter. In
figure 7.3 chord CD is the diameter. It can be noticed immediately that the diameter is twice
the radius of the circle. The center of the circle is the mid point of the diameter. A circle has
infinite diameters and all have the same length.
Example 1
A, B, C & D lie on a circle with center P. Classify the following segments as radii and
chords.
PA, AB, AC, BP, DP, DA, PC, BC, BD, CD.
Solution:
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PinkMonkey.com Geometry Study Guide - 7.2 Lines of a Circle
Example 2
Name the secant and the tangent in the following figure :
Solution:
secant - l
tangent - m
Example 3
P is the center of a circle with radius 5 cm. Find the length of the longest chord of the circle.
Solution:
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PinkMonkey.com Geometry Study Guide - 7.3 Arcs
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Index
7.3 Arcs
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The angle described by any two radii of a circle is called the central angle. Its vertex is the
center of the circle. In figure 8.4 ∠ APB is a central angle. The part of the circle that is cut
by the arms of the central angle is called an arc. AB is an arc and so is AOB . They are
represented as
&
.
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Figure 7.4
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is called the minor arc and
is the major arc. The minor arc is always represented
by using the two end points of the arc on the circle. However it is customary to denote the
major arc using three points. The two end points of the major arc and a third point also on
the arc. If a circle is cut into two arcs such that there is no minor or major arc but both the
arcs are equal then each arc is called a semicircle.
An arc is measured as an angle in degrees and also in units of length. The measure of the
angle of an arc is its central angle and the length of the arc is the length of the portion of the
circumference that it describes.
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7.1 Introduction
7.2 Lines of circle
7.3 Arcs
7.4 Inscribed
angels
7.5 Some
properties od
tangents,
secants and
chords
7.6 Chords and
their arcs
7.7 Segments of
chords secants
and tangents
7.8 Lengths of
arcs and area of
sectors
Chapter 8
PinkMonkey.com Geometry Study Guide - 7.3 Arcs
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angle of an arc AB = m
length of an arc AB = l
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Since the measure of the angle of an arc is its central angle, if two central angles have equal
measure then the corresponding minor arcs are equal.
Conversely if two minor arcs have equal measure then their corresponding central angles are
equal.
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Index
7.4 Inscribed angles
Whereas central angles are formed by radii, inscribed angles are formed by chords. As
shown in figure 8.5 the vertex o of the inscribed angle AOB is on the circle. The minor arc
cut on the circle by an inscribed angle is called as the intercepted arc.
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Figure 7.5
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Theorem: The measure of an inscribed angle is half the measure of its intercepted arc.
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Proof: For a circle with center O ∠ BAC is the inscribed angle and arc BXC is the
intercepted arc. To prove that m ∠ BAC = 1/2 m (arc BXC). There arise three cases as
shown in figure 8.6 (a), 8.6 (b) and 8.6 (c).
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7.1 Introduction
7.2 Lines of circle
7.3 Arcs
7.4 Inscribed
angels
7.5 Some
properties od
tangents,
secants and
chords
7.6 Chords and
their arcs
7.7 Segments of
chords secants
and tangents
7.8 Lengths of
arcs and area of
sectors
Chapter 8
PinkMonkey.com Geometry Study Guide - 7.5 Some properties of tangents, secants and chords
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Index
7.5 Some properties of tangents, secants and chords
Theorem: If the tangent to a circle and the radius of the circle intersect they do so at right
angles :
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Figure 7.9 (a)
Figure 7.9 (b)
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In figure 7.9 (a) l is a tangent to the circle at A and PA is the radius.
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To prove that PA is perpendicular to l , assume that it is not.
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Now, with reference to figure 7.9 (b) drop a perpendicular from P onto l at say B. Let D be a
point on l such that B is the midpoint of AD.
In figure 7.9 (b) consider ∆ PDB and ∆ PAB
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7.1 Introduction
7.2 Lines of circle
7.3 Arcs
7.4 Inscribed
angels
7.5 Some
properties od
tangents,
secants and
chords
7.6 Chords and
their arcs
7.7 Segments of
chords secants
and tangents
7.8 Lengths of
arcs and area of
sectors
Chapter 8
PinkMonkey.com Geometry Study Guide - 7.5 Some properties of tangents, secants and chords
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seg.BD ≅ seg.BA ( B is the midpoint of AD)
∠PBD ≅ ∠PBA ( PB is perpendicular to l ) and
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seg.PB = seg.PB (same segment)
∴ ∆ PBD ≅ ∆ PAB (SAS)
∴ seg.PD ≅ seg.PA corresponding sides of congruent triangles are congruent.
∴ D is definitely a point on the circle because l (seg.PD) = radius.
D is also on l which is the tangent. Thus l intersects the circle at two distinct points A and D.
This contradicts the definition of a tangent.
Hence the assumption that PA is not perpendicular to l is false. Therefore PA is
perpendicular to l.
Angles formed by intersecting chords, tangent and chord and two secants: If two chords
intersect in a circle, the angle they form is half the sum of the intercepted arcs.
In the figure 7.10 two chords AB and CD intersect at E to form ∠1 and ∠ 2.
Figure 7.10
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PinkMonkey.com Geometry Study Guide - 7.5 Some properties of tangents, secants and chords
m ∠1 =
(m seg.AD + m seg.BC) and
m ∠2 =
(m seg.BD + m seg.AC)
Tangent Secant Theorem: If a chord intersects the tangent at the point of tangency, the
angle it forms is half the measure of the intercepted arc. In the figure 7.11 l is tangent to the
circle. Seg.AB which is a chord, intersects it at B which is the point of tangency.
Figure 7.11
The angles formed ∠ ABX and ∠ ABY are half the measures of the arcs they intercept.
m∠1=
m (arc ACB)
m∠2=
m (arc AB)
This can be proved by considering the three following cases.
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PinkMonkey.com Geometry Study Guide - 7.6 Chords and their arcs
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Index
7.6 Chords and their arcs
Theorem: If in any circle two chords are equal in length then the measures of their
corresponding minor arcs are same.
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As shown in figure 8.16 AB and CD are congruent chords. Therefore according to the
theorem stated above m (arc AB) = m (arc CD) or m ∠ AOB = m ∠ COD.
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Figure 7.16
To prove this, join A, B, C & D withO.
Consider ∆ AOB and ∆ COD
seg.AO ≅ seg.CO and seg.OB ≅ seg.OD (radii of a circle are always congruent).
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7.1 Introduction
7.2 Lines of circle
7.3 Arcs
7.4 Inscribed
angels
7.5 Some
properties od
tangents,
secants and
chords
7.6 Chords and
their arcs
7.7 Segments of
chords secants
and tangents
7.8 Lengths of
arcs and area of
sectors
seg.AB ≅ seg.CD (given)
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PinkMonkey.com Geometry Study Guide - 7.6 Chords and their arcs
Theorem: The perpendicular from the center of a circle to a chord of the circle bisects the
chord.
Figure 7.18
In figure 7.18, XY is the chord of a circle with center O. Seg.OP is the perpendicular from
the center to the chord. According to the theorem given above seg XP = seg. YP.
To prove this, join OX and OY
Consider ∆ OXP and ∆ OXY
Both are right triangles.
hypotenuse seg. OX ≅ hypotenuse seg. OY
( both are radii of the circle )
seg. OP ≅ seg OP (same side)
∴ ∆ OXP ≅ ∆ OYP (H.S.)
∴ seg XP ≅ seg. YP (corresponding sides of congruent triangles are congruent).
∴ P is the midpoint of seg.XY.
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PinkMonkey.com Geometry Study Guide - 7.7 Segments of chords secants and tangents
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Index
7.7 Segments of chords secants and tangents
Theorem : If two chords, seg.AB and seg.CD intersect inside or outside a circle at P then l
(seg. PA) × l (seg. PB) = l (seg. PC) × l (seg. PD)
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Figure 7.21 (a)
Figure 7.21 (b)
In figure 7.21 (a) P is in the interior of the circle. Join AC and BD and consider ∆APC and
∆BDP.
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m ∠ APC = m ∠ BPD (vertical angles).
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m ∠ CAP = m ∠ BDP (angles inscribed in the same arc).
∴ ∆APC ∼ ∆ BPD ( A A test )
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7.1 Introduction
7.2 Lines of circle
7.3 Arcs
7.4 Inscribed
angels
7.5 Some
properties od
tangents,
secants and
chords
7.6 Chords and
their arcs
7.7 Segments
of chords
secants and
tangents
7.8 Lengths of
arcs and area of
sectors
Chapter 8
PinkMonkey.com Geometry Study Guide - 7.7 Segments of chords secants and tangents
∴ (corresponding sides of similar triangles).
∴ l (seg. PA) × l (seg. PB) = l (seg. PT)2.
Theorem: The lengths of two tangent segments from an external point to a circle are equal.
As shown in figure 7.23 seg. QR and seg. QS are two tangents on a circle with P as its center.
Click here to enlarge
Figure 7.23
To prove that l (seg.QR) = l (seg.QS) join P to Q and R to S.
m ∠ PRQ = m ∠ PSQ = 900.
The radius and the tangent form a right angle at the point of tangency,
∴ ∆ PRQ and ∆ PSQ are right triangles such that
seg. PR ≅ seg PS (radii of the same circle).
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PinkMonkey.com Geometry Study Guide - 7.8 Lengths of arcs and areas of sectors
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Index
7.8 Lengths of arcs and areas of sectors
An arc is a part of the circumference of the circle; a part proportional to the central angle.
Home
If 3600 corresponds to the full circumference. i.e. 2 π r then for a central angle of x0 (figure
7.24) the corresponding arc length will be l such that
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Figure 7.24
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Analogically consider the area of a sector. This too is proportional to the central angle. 3600
corresponds to area of the circle π r2. Therefore for a central angle m0 the area of the sector
will be in the ratio :
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7.1 Introduction
7.2 Lines of circle
7.3 Arcs
7.4 Inscribed
angels
7.5 Some
properties od
tangents,
secants and
chords
7.6 Chords and
their arcs
7.7 Segments of
chords secants
and tangents
7.8 Lengths of
arcs and area
of sectors
Chapter 8
PinkMonkey.com Geometry Study Guide - 7.4 Inscribed angles
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Index
Thus it is proved that the measure of the inscribed angle is half that of the intercepted arc.
Theorem: If two inscribed angles intercept the same arc or arcs of equal measure then the
inscribed angles have equal measure.
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Figure 7.7
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In figure 7.7 ∠ CAB and ∠ CDB intercept the same arc CXB.
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Prove that m ∠ CAB = ∠ CDB.
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From the previous theorem it is known that
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m ∠ CAB =
m (arc CXB) and also
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7.1 Introduction
7.2 Lines of circle
7.3 Arcs
7.4 Inscribed
angels
7.5 Some
properties od
tangents,
secants and
chords
7.6 Chords and
their arcs
7.7 Segments of
chords secants
and tangents
7.8 Lengths of
arcs and area of
sectors
Chapter 8
PinkMonkey.com Geometry Study Guide - 7.4 Inscribed angles
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m ∠ CDB =
m (arc CXB)
∴ m ∠ CAB = m ∠ CDB
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Therefore if two inscribed angles intercept the same arc or arcs of equal measure the two
inscribed angles are equal in measure.
Theorem: If the inscribed angle intercepts a semicircle the inscribed angle measures 900.
Figure 8.8
The inscribed angle ∠ ACB intercepts a semicircle arc AXB (figure 8.8). We have to prove
that m ∠ ACB = 900.
m ∠ ACB =
=
m (arc AXB)
(1800)
= 900
Therefore if an inscribed angle intercepts a semicircle the inscribed angle is a right angle.
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PinkMonkey.com Geometry Study Guide - 7.4 Inscribed angles
Example 1
a) In the above figure name the central angle of arc AB.
b) In the above figure what is the measure of arc AB.
c) Name the major arc in the above figure.
Solution:
a) ∠ AOB
b) 800. The measure of an arc is the measure of its central angle.
c) Arc AXB
Example 2
a) In the above figure name the inscribed angle and the intercepted arc.
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PinkMonkey.com Geometry Study Guide - 7.4 Inscribed angles
b) What is m (arc PQ)
Solution:
a) inscribed angle - ∠ PRQ
intercepted arc - arc PQ
b) 600. The measure of an intercepted arc is twice the measure of its inscribed angle.
Example 3
Ð PAQ and ∠ PBQ intercept the same arc PQ what is the m ∠ PBQ and m (arc PQ) ?
Solution:
m ∠ PBQ = 400 If two inscribed angles intercept the same arc their measures are equal m
(arc PQ) = 800 as m (arc) = 2m (inscribed angle).
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PinkMonkey.com Geometry Study Guide - 7.6 Chords and their arcs
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Figure 7.19
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Thus if two chords are equal in measure they are equidistant from the center of the circle.
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The converse of this theorem is that if two chords are equidistant from the center of the
circle, they are equal in measure.
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As shown in figure 7.20 if seg. HI and seg. JK are two chords equidistant from the center of
the circle, they are equal in length.
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7.1 Introduction
7.2 Lines of circle
7.3 Arcs
7.4 Inscribed
angels
7.5 Some
properties od
tangents,
secants and
chords
7.6 Chords and
their arcs
7.7 Segments of
chords secants
and tangents
7.8 Lengths of
arcs and area of
sectors
Chapter 8
PinkMonkey.com Geometry Study Guide - 7.6 Chords and their arcs
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Figure 7.20
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To prove that seg.HI ≅ seg.JK join OI and OK.
Survey
Consider ∆ OIP and ∆ OKQ, ( both are right triangles) .
seg.OI ≅ seg.OK, ( both are radii of the same circle).
seg.OP ≅ seg.OQ (given that chords are equidistant from the center O).
∴ ∆ OIP ≅ ∆ OKQ (H.S.)
∴ seg.PI ≅ seg.QK (corresponding sides of congruent triangles are congruent).
Also it is known that the perpendicular from the center bisects the chord. Therefore, seg. HI
≅ seg JK.
Example 1
AB and CD are chords in a circle with center O. l (seg.AB ) = l (seg.CD) = 3.5 cm and m ∠
COD = 950. Find m arc AB.
Solution:
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PinkMonkey.com Geometry Study Guide - 7.6 Chords and their arcs
950
m arc AB = m ∠ AOB
Since ∆ AOB ≅ ∆ COD by SSS m ∠ AOB = m ∠ COD.
Example 2
PQ is a chord of a circle with center O. Seg.OR is a radius intersecting PQ at right angles at
point T. If l (PT) = 1.5 cm and m arc PQ = 800, find l (PQ) and m arc PR.
Solution:
l (PQ) = 3
m (arc PR) = 400
Seg.OT is perpendicular to PQ and therefore bisects PQ at T.
∴ l (.PQ) = 2 l (PT)
Seg.OR bisects arc PQ. ∴ m (arc PR) =
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m (arc PQ)
PinkMonkey.com Geometry Study Guide - 7.6 Chords and their arcs
Example 3
Seg HI and seg. JK are chords of equal measure in a circle with center O. If the distance
between O and seg. HI is 10 cm find the length of the perpendicular from O onto seg.JK.
Solution:
10 cm.
Chords of equal measure are equidistant from the center.
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Index
Example 1
Two chords seg AB and seg. CD intersect in the circle at P. Given that l (seg.PC) = l (seg.
PB) = 1.5 cm and l (seg.PD) = 3 cm. Find l (seg.AP).
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Solution:
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l (seg AP) × l (seg PB) = l (seg DP) × l (seg PC)
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l (seg AP) × 1.5 = 3 × 1.5
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7.1 Introduction
7.2 Lines of circle
7.3 Arcs
7.4 Inscribed
angels
7.5 Some
properties od
tangents,
secants and
chords
7.6 Chords and
their arcs
7.7 Segments
of chords
secants and
tangents
7.8 Lengths of
arcs and area of
sectors
l (seg AP) = 3 cm.
Example 2
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PinkMonkey.com Geometry Study Guide - 7.7 Segments of chords secants and tangents
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Seg.PA and seg.PC are two secants where l (seg.PB) = 3.5 cm, l (seg.PD) = 4 cm and l (seg.
DC) = 3 cm. Find l (seg.AB).
Solution:
l (seg AP) × l (seg BP) = l (seg CP) × l (seg DP)
x × 3.5 = 7 × 4
x=8
l (seg AB) = l (seg AP) − l (seg BP)
= 8 - 3.5
= 4.5 cm.
Example 3
PT is a tangent intersecting the secant through AB at P. Given l (seg. PA) = 2.5 cm. and l
(seg.AB) = 4.5 cm., find l (seg PT).
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PinkMonkey.com Geometry Study Guide - 7.7 Segments of chords secants and tangents
Solution:
l (seg PT)2 = l (seg PA) × l (seg PB)
= 2.5 × 7
= 17.5
l (seg. PT) = 4.2 cm.
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PinkMonkey.com Geometry Study Guide - 7.5 Some properties of tangents, secants and chords
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In figure 7.15 l and m are secants. l and m intersect at O outside the circle. The intercepted
and
.
arcs are
∠ COD = ( m
-m
)
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Figure 7.15
Test Prep
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Help / FAQ
How to Cite
Conclusion :
(a) If two chords intersect in a circle the angle formed is half the sum of the measures of the
intercepted arcs.
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(b) Angle formed by a tangent and a chord intersecting at the point of tangency is half the
measure of the intercepted arcs.
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(c) Angle formed by two secants intersecting outside the circle is half the difference of the
measures of the intercepted arcs.
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Index
7.1 Introduction
7.2 Lines of circle
7.3 Arcs
7.4 Inscribed
angels
7.5 Some
properties od
tangents,
secants and
chords
7.6 Chords and
their arcs
7.7 Segments of
chords secants
and tangents
7.8 Lengths of
arcs and area of
sectors
Chapter 8
PinkMonkey.com Geometry Study Guide - 7.5 Some properties of tangents, secants and chords
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Example 1
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In the above figure seg.AB and seg.CD are two chords intersecting at X such that m ∠ AXD
= 1150 and m (arc CB) = 450 . Find m arc APD.
Solution:
m arc APD = 1850
m ∠ AXD =
{ m (arc APD) + m (arc CB) }
m ( arc APD) = 2 m ∠ AXD - m (arc CB)
= 2 × 1150 − 450
= 1850
Example 2
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PinkMonkey.com Geometry Study Guide - 7.5 Some properties of tangents, secants and chords
l is a tangent to the circle at B. Seg. AB is a chord such that m ∠ ABC = 500. Find the m (arc
AB).
Solution:
m (arc AB) = 1000
m ∠ ABC =
50 =
m (arc AB)
m (arc AB)
m (arc AB) = 1000
Example 3
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PinkMonkey.com Geometry Study Guide - 7.5 Some properties of tangents, secants and chords
l and m are secants to the circle intersecting each other at A. The intercepted arcs are arc PQ
and arc RS if m ∠ PAQ = 250 and m ∠ ROS = 800 find m (arc PQ).
Solution:
m ( arc PQ) = 300
m ∠ PAQ =
{ m (arc RS) - m (arc PQ)
2 m ∠ PAQ = m (arc RS) - m (arc PQ)
\ m (arc PQ) = m (arc RS) - 2 m ∠ PAQ
= 800 - 500
= 300
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