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Question

AB is diameter and AC is a chord of a circle such that BAC=30. If the tangent at C intersects AB produced in D, prove that BC = BD [4 MARKS]

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Solution

Concept: 1 Mark
Application: 3 Marks

Solution:

As OA = OC (radius)

OAC=OCA=30

As OC = OB

OCB=OBC

In BAC,

OAC+OCA+OCB+OBC=180

30+30+2OCB=180

OCB=60

In BOC,

OCB+OBC+BOC=180

60+60+BOC=180

BOC=60

As the tangent at any point of a circle is perpendicular to the radius through the point of contact

OCD=90

BCD=OCDBCO

BCD=9060

BCD=30 …… (i)

In OCD,

OCD+ODC+DOC=180

90+ODC+60=180

ODC=30 ….. (ii)

From (i) and (ii)

BC = BD


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