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Tangents and Intersecting Chords — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsTangents and Intersecting Chords4 Marks
Question
In a cyclic quadrilateral ABCD, the diagonal AC bisects the angle BCD. Prove that the diagonal BD is parallel to the tangent to the circle at point A.
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Answer


$\angle ADB =\angle ACB$.....(i) (Angles in same segement)
Similarly ,
$
\angle ABD =\angle ACD....(ii)$
But, $\angle ACB =\angle ACD ( AC$ is bisector of $\angle BCD )$
$\therefore \angle A D B=\angle A B D$ (from (i) and (ii))
TAS is a tangent and $A B$ is a chord
$\therefore \angle BAS =\angle ADB$ (angles in alternate segment)
But, $\angle ADB =\angle ABD$
$
\therefore \angle BAS =\angle ABD$
But these are alternate angles
Therefore, TS || BD

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