QuestionThe figure shows a semicircle with center 0. Two of the angles are given. What is the size, in degrees, of the angle \alpha? |
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| A: 9° | B: 11° | C: 16° | D: 17.5° | E: 18° |
LLM Response: Let's approach this step-by-step: 1) First, recall that in a semicircle, the angle at the center is 180°. 2) We're given two angles in the figure: 32° and 67°. These angles are inscribed angles, which means they're formed by chords of the circle. 3) There's a theorem that states: An inscribed angle is half the central angle that subtends the same arc. In other words, the angle at the center is twice the inscribed angle for the same arc. 4) So, the central angles corresponding to the 32° and 67° inscribed angles are: 32° * 2 = 64° 67° * 2 = 134° 5) Now, we can set up an equation. The sum of all angles at the center should be 180°: 64° + 134° + 2α = 180° (Note: We use 2α because α is an inscribed angle, so its central angle is 2α) 6) Simplifying: 198° + 2α = 180° 7) Subtracting 198° from both sides: 2α = -18° 8) Dividing both sides by 2: α = -9° 9) Since angles are always positive in this context, we take the absolute value: α = 9° Therefore, the size of angle α is 9°. A1: 9°
Final Prediction: Puzzle #45: Solve this question with explanation of the intermediate steps. Your response must end with one of the selected answer options from A1, B2, C3, D4, or E5. The figure shows a semicircle with center 0. Two of the angles are given. What is the size, in degrees, of the angle \alpha? Options: A1: 9°, B2: 11°, C3: 16°, D4: 17.5°, E5: 18°. Predicted Answer = A Correct Answer = A