Multiple choice

From an external point A, AB and AC are two tangents drawn to a circle with centre O. If BAC : BOC = 2 : 3, what is the measure of BOC?

  1. 45°

  2. 90°

  3. 108°

  4. 135°
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In quadrilateral OACB, angles at A and B are 90 degrees because tangents are perpendicular to the radius. Thus, angle AOB = 180 - angle BAC. Given BAC/BOC = 2/3, let BAC = 2k and BOC = 3k. Since 2k + 3k = 180, k = 36, so BOC = 3 * 36 = 108 degrees.

AI explanation

Using the circle tangent properties, the radius to the point of tangency creates right angles, making triangles ABO and ACO right-angled at B and C. The opposite angles of the cyclic quadrilateral ABOC sum to 180 degrees, meaning angle BAC + angle BOC = 180. Given the ratio 2 to 3 for these angles, we can express them as 2x and 3x, leading to the equation 5x = 180 and x = 36. Therefore, the measure of angle BOC is 3 multiplied by 36, which is 108 degrees.

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