Multiple choice

Draw a circle of radius $3\ cm$. Take a point outside the circle at a distance of $6\ cm$ from the centre of the circle. Construct tangents from this point to the circle. Measure the angle between the tangents and select the correct value from below.

  1. $60^\circ$
  2. $30^\circ$
  3. $45^\circ$
  4. $90^\circ$
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A Correct answer
Explanation

In the right-angled triangle formed by the center of the circle, the point of tangency, and the external point, the sine of half the angle between the tangents is equal to the radius divided by the distance to the center, which is 3/6 = 1/2. This means half the angle is 30 degrees, so the total angle between the tangents is 60 degrees.

AI explanation

The tangent at any point on a circle is perpendicular to the radius drawn to that point, forming right angled triangles with the center, the external point, and the point of tangency. In the right triangle formed by the center, the external point, and a point of tangency, the hypotenuse is the distance from the center to the external point, which is 6 cm, and the side opposite the angle at the external point is the radius of 3 cm. Using the trigonometric ratio for sine, sin(angle) equals 3 divided by 6, which equals 0.5. The inverse sine of 0.5 is 30 degrees, which is the angle between the tangent and the line connecting the external point to the center. Because the line from the center to the external point bisects the angle between the two tangents, the total angle between the tangents is 2 times 30 degrees. Therefore, the measured angle between the tangents is 60 degrees.

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