Question

Show that the tangents drawn at the ends of a chord subtend congruent angles with the chord.

Answer 1

The diagram $\Delta AOC\&\Delta OBC$

$OA=OB$(radius of circle)

$OC=OC$(common)

$\angle OAC$=$\angle OBC$(tangent subtend right angle with the radius) so

SSA

$\Delta AOC\cong \Delta OBC$

So $AC=BC$ & $\angle ACP=\angle PCB$

Now in $\Delta APC\&\Delta BPC$

$AC=BC$(proved)

$\angle ACP=\angle PCB$(proved)

$PC=PC=$common

so $\angle PAC=\angle PBC$ (proved)

So tangent drawn at ends of chord subtend right angle with the chord.