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In figure 7.15 l and m are secants. l and m intersect at O outside the circle. The intercepted arcs are and .

Ð COD = ( m - m )

Figure 7.15

Conclusion :

(a) If two chords intersect in a circle the angle formed is half the sum of the measures of the intercepted arcs.

(b) Angle formed by a tangent and a chord intersecting at the point of tangency is half the measure of the intercepted arcs.

(c) Angle formed by two secants intersecting outside the circle is half the difference of the measures of the intercepted arcs.

Example 1

In the above figure seg.AB and seg.CD are two chords intersecting at X such that m Ð AXD = 1150 and m (arc CB) = 450 . Find m arc APD.

Solution:

m arc APD = 1850

m Ð AXD = { m (arc APD) + m (arc CB) }

m ( arc APD) = 2 m Ð AXD - m (arc CB)

= 2 ´ 1150 - 450

= 1850

Example 2

l is a tangent to the circle at B. Seg. AB is a chord such that m Ð ABC = 500. Find the m (arc AB).

Solution:

m (arc AB) = 1000

m Ð ABC = m (arc AB)

50 = m (arc AB)

m (arc AB) = 1000

Example 3

l and m are secants to the circle intersecting each other at A. The intercepted arcs are arc PQ and arc RS if m Ð PAQ = 250 and m Ð ROS = 800 find m (arc PQ).

Solution:

m ( arc PQ) = 300

m Ð PAQ = { m (arc RS) - m (arc PQ)

2 m Ð PAQ = m (arc RS) - m (arc PQ)

\ m (arc PQ) = m (arc RS) - 2 m Ð PAQ

= 800 - 500

= 300

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

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