Multiple choice

$\Delta ABC$ is inscribed in a circle with center $O$. If $\angle OAB = { 40 }^{\circ}$, then $\angle ACB$ is equal to

  1. ${ 50 }^{\circ} $
  2. ${ 40 }^{\circ} $
  3. ${ 60 }^{\circ} $
  4. ${ 70 }^{\circ} $
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A Correct answer
Explanation

In triangle OAB, OA = OB (radii), so angle OBA = angle OAB = 40 degrees. Angle AOB = 180 - (40 + 40) = 100 degrees. The angle subtended by arc AB at the center is 100 degrees. The angle at the circumference (angle ACB) is half of the angle at the center, so 100 / 2 = 50 degrees.

AI explanation

Because OA and OB are both radii of the circle, triangle OAB is isosceles, which means angle OBA is also 40 degrees. The third angle in the triangle, angle AOB, is 180 degrees minus 40 degrees minus 40 degrees, equaling 100 degrees. The central angle AOB and the inscribed angle ACB subtend the same arc AB. Therefore, angle ACB is half of angle AOB, which gives the final result of 50 degrees.

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