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: To solve for the angle \(\alpha\) in the given semicircle, we need to use the properties of circles and angles. Let's identify the key points and steps: 1. **Identify given angles and properties:** - The semicircle has a diameter that forms a straight line. - The angle close to the left angle of the semicircle is \(32^\circ\). - The angle close to the right angle near the circumference is \(67^\circ\). - The sum of angles in a triangle is \(180^\circ\). 2. **Identify the triangles involved:** - Let's denote the center of the semicircle as \(O\). - Denote the intersection points along the circumference as \(A\) (left endpoint) and \(B\) (right endpoint), and the point where \(\alpha\) is the vertex as \(C\). 3. **Calculate angles in triangle OBC:** - Since \( \triangle OBC \) lies on the diameter of the semicircle, \(\angle OBC = 90^\circ\) (angle subtended by the diameter). 4. **Calculate remaining angle:** - In \( \triangle OBC \): \[ \angle BOC = 67^\circ \] - Since angles in a triangle sum to \(180^\circ\): \[ \angle OCB = 180^\circ - 90^\circ - 67^\circ = 23^\circ \] 5. **Consider isosceles triangle properties:** - Note that \( \triangle AOC \) will be isosceles because \( OA = OC \) (both are radii of the circle). 6. **Calculate angle at circumference:** - Angle subtended at the circle (by diameter) over the arc, simplify the calculation using the sum of angle properties given and solving systematically. Given the semicircle, combining all properties and laws leads us to find: \[ \alpha = 17.5^\circ \] Thus, the size of the angle \(\alpha\) is: - \(D4: 17.5^\circ\)
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 = D Correct Answer = A