Multiple choice

A triangle with vertices A, B, C is inscribed inside a circle such that the longest side of the triangle$(BC)$ is also the diameter of the circle. If $\angle$ABC is 30$^{\circ}$ and the length of AC is 5cm. The diameter of the circle is:

  1. 5$\sqrt{3}$

  2. 8

  3. 10
  4. 5$\sqrt{5}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In a right triangle inscribed in a circle with the hypotenuse as the diameter, the angle opposite the diameter is 90 degrees. Given angle ABC = 30 degrees, the triangle is a 30-60-90 triangle. AC is opposite the 30-degree angle. AC = Diameter * sin(30) = Diameter * 0.5. 5 = Diameter * 0.5, so Diameter = 10.

AI explanation

Because BC is the diameter, angle BAC equals 90 degrees by Thales's theorem. In right triangle ABC, we use the sine ratio: sin 30 degrees equals the opposite side AC divided by the hypotenuse BC. So, 0.5 equals 5 divided by BC, which gives a diameter BC of 10 cm.

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