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МатематикаВопрос:
Analyze the geometry problem: 'Дано: AB и AC — касательные; ∠A = 60°. Найти: ∠BOC.'. In this problem, AB and AC are tangents to a circle with center O, from an external point A. The angle ∠A (which is ∠BAC) is given as 60°. The task is to find the angle ∠BOC.
Ответ:
Краткая запись:
- Дано: AB и AC — касательные к окружности с центром O. ∠BAC = 60°.
- Найти: ∠BOC.
Краткое пояснение: Четырехугольник ABOC образован двумя радиусами (OB и OC) и двумя касательными (AB и AC). Углы при точках касания (∠ABO и ∠ACO) являются прямыми (90°), так как радиус перпендикулярен касательной. Сумма углов четырехугольника равна 360°.
Пошаговое решение:
- Шаг 1: Так как AB и AC — касательные, а OB и OC — радиусы, проведенные к точкам касания, то ∠ABO = 90° и ∠ACO = 90°.
- Шаг 2: Рассмотрим четырехугольник ABOC. Сумма всех его внутренних углов равна 360°.
- Шаг 3: Углы четырехугольника: ∠BAC, ∠ABO, ∠BOC, ∠ACO.
- Шаг 4: Подставляем известные значения: 60° (∠BAC) + 90° (∠ABO) + ∠BOC + 90° (∠ACO) = 360°.
- Шаг 5: Складываем известные углы: 60° + 180° + ∠BOC = 360°.
- Шаг 6: Упрощаем: 240° + ∠BOC = 360°.
- Шаг 7: Находим ∠BOC: ∠BOC = 360° - 240° = 120°.
Ответ: ∠BOC = 120°.
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