Question

In the figure above (not to scale) AB and AC are two tangents drawn to a circle at B and C respectively $\angle DCA={35}^{\circ }$ and $\angle DBA={40}^{\circ }.$ Find the measure of $\angle BAC.$

• A. ${15}^{\circ }$
• B. ${30}^{\circ }$
• C. ${45}^{\circ }$
• D. ${60}^{\circ }$

Answer 1

The correct option is B ${30}^{\circ }$

Given

$\angle DBA={40}^{\circ },\angle DCA={35}^{\circ }$

According to the alternate segment theorem, angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

Since $AC$ and $AB$ are tangents.

$\angle ACO=\angle CBO,\angle ABD=\angle BCD$

$⟹\angle CBD={35}^{\circ },\angle BCD={40}^{\circ }$

In $△ABC$

$\angle ABC+\angle BCA+\angle CAB=180$

$(40+35)+(35+40)+\angle BAC=180$

$\angle BAC=180–150={30}^{\circ }$