Question

△ABC is inscribed in a circle. Find the angle between the tangents to the circle at points B and C, if m∠CAB=50°.

Answer 1

Answer:

80°

Step-by-step explanation:

We know the measure of ∠CAB.  This is an inscribed angle; this means its measure is 1/2 that of the intercepted arc, BC.  This means the measure of BC is 100°.

This makes the measure of arc BAC is 360-100 = 260°.

The measure of the angle formed by the tangents, since it is outside the circle, will be 1/2 of the difference of the intercepted arcs.  This means the measure of this angle will be

1/2(260-100) = 1/2(160) = 80°

Answer 2

$DB,DC \: are \: tangents \: to \: (O) \: at \: B,C\\\Rightarrow DB\perp OB \: at \: B, DC\perp OC \: at \: C\Rightarrow \widehat{DBO}=\widehat{DCO}=90° \\ \widehat{BOC}=2\widehat{BAC}=2\times 50°=100°\\\widehat{BDC}=360°-\widehat{BOC}-\widehat{DBO}-\widehat{DCO}=360°-100°-90°-90°=80°$