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Arc AXB can be

a) Semi circle

b) minor arc

c) major arc

Case 1 : Assume arc AXB is a semicircle when Ð ABC intercepts a semicircle the chord AB passes through the center. Therefore m Ð ABC = 900 (a tangent is always perpendicular to the diameter that intersects it at the point of tangency).

m (arc BXA) = 1800 (arc BXA is a semi circle)

\ m (arc BXA) = ´ 1800 = 900

\ m Ð ABC = m (arc BXA)

Case 2 : Assume that ÐABC intercepts a minor arc. Therefore as seen in figure 8.13 the center O lies in the exterior of Ð ABC.

Figure 7.13

m Ð ABC = 900 - m Ð ABO

m Ð ABO = 900 - m Ð ABC ----------- (1)

But m Ð ABO = m Ð OAB

(as OAB is an isosceles triangle )

\ m Ð OAB = 900 - m Ð ABC ----------- (2)

(1) + (2)

\ m Ð ABO + m Ð OAB = 180 - 2 m ÐABC

Since m ÐABO + m Ð OAB = 180 - m Ð BOA

180 - m Ð BOA = 180 - 2 m Ð ABC

i.e. m Ð BOA = 2 m Ð ABC

m Ð ABC = m Ð BOA

m Ð ABC = m ( arc AXB )

Case 3 :

Figure 7.14

If Ð ABC intercepts a major arc, the center of the circle O will lie in the interior as ÐABC . See figure 7.14.

Now ÐADB intercepts a minor arc AYB.

\ m Ð AOB = m (arc AYB)

\ 1800 - m ÐADB = { 3600 - m (arc AXB) }

\ 1800 - m ÐADB = 1800 - m (arc AXB)

\ m Ð ADB = m (arc AXB)

If two secants intersect outside a circle half the difference in the measures of the intercepted arcs gives the angle formed by the two secants.

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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