2. Математический диктант. Дано: ∪ACB : ∪ADB = 3:5 Найти: ∠BAE

Решение:

• Пусть дуга ACB равна 3x, а дуга ADB равна 5x.
• Сумма этих дуг составляет полную окружность: 3x + 5x = 360°.
• 8x = 360°.
• x = 360° / 8 = 45°.
• Дуга ACB = 3 * 45° = 135°.
• Дуга ADB = 5 * 45° = 225°.
• Угол BAE является вписанным углом, опирающимся на дугу BE.
• Для нахождения дуги BE, нам нужно знать положение точки E. Из рисунка видно, что E лежит на продолжении диаметра, проходящего через A. Линия AE является касательной к окружности в точке A, или секущей. По рисунку AE - это луч, исходящий из A.
• Если AE - касательная, то ∠BAE опирается на дугу AB.
• Дуга AB = ∪ADB - ∪AD (или ∪ACB - ∪CB).
• Из рисунка, дуга AD = 52°, дуга BC = 70°.
• Это из другого задания. Вернемся к данному.
• В данном задании мы имеем: ∪ACB = 135°, ∪ADB = 225°.
• Угол BAE, если AE - касательная, опирается на дугу AB.
• Дуга AB = ∪ADB - ∪AD. Нам неизвестна дуга AD.
• Другой вариант: ∠BAE - это угол между хордой AB и касательной AE. Он равен половине дуги AB.
• Нужно найти дугу AB.
• Если ACВ = 135°, то дуга AB = 360° - 135° = 225°. Это неверно, т.к. ACB - это дуга, а не центральный угол.
• Let's rethink. ∪ACB and ∪ADB are arcs. They add up to 360°.
• ∪ACB = 135°, ∪ADB = 225°.
• ∠BAE is the angle we need to find. It's an inscribed angle if E is on the circle, but E is outside.
• It's likely that AE is a tangent line at A. In that case, the angle formed by the tangent AE and the chord AB is equal to half the measure of the arc AB subtended by the angle.
• We need to find the measure of arc AB.
• We know ∪ACB = 135° and ∪ADB = 225°.
• These notations usually mean the arc that doesn't contain the third point. So ∪ACB means the arc from A to B going through C, and ∪ADB means the arc from A to B going through D. This is confusing.
• Let's assume ∪AC + ∪CB = 135° and ∪AD + ∪DB = 225°. This doesn't help directly.
• Let's assume ∪AB (minor arc) = A and ∪BA (major arc) = B. Then A+B = 360.
• The notation ∪ACB refers to the arc from A to B going counterclockwise (or clockwise, depending on convention, but usually counterclockwise). The notation ∪ADB refers to the arc from A to B going in the opposite direction.
• So, let ∪AB (minor) = y. Then the major arc AB = 360° - y.
• Let's assume ∪ACB = 135° means the arc from A to B that contains C.
• Let's assume ∪ADB = 225° means the arc from A to B that contains D.
• So, the minor arc AB is 225° or 135°. This is contradictory.
• Let's interpret ∪ACB as the arc measured from A to B through C, and ∪ADB as the arc from A to B through D.
• This means these are two different arcs connecting A and B.
• Let the measure of the minor arc AB be 'a' and the measure of the major arc AB be 'b'. a + b = 360°.
• If ∪ACB : ∪ADB = 3:5, it means the ratio of the two arcs between A and B is 3:5.
• So, let the arcs be 3x and 5x.
• 3x + 5x = 360°
• 8x = 360°
• x = 45°.
• The two arcs between A and B are 3 * 45° = 135° and 5 * 45° = 225°.
• So, one arc AB is 135° and the other is 225°.
• We need to find ∠BAE. AE is a line passing through A and E. From the diagram, AE is a tangent to the circle at A.
• The angle ∠BAE is formed by the tangent AE and the chord AB.
• This angle is equal to half the measure of the arc AB intercepted by the angle. The intercepted arc is the minor arc AB.
• We need to determine which is the minor arc. Let's assume the minor arc AB is 135°.
• Then ∠BAE = 135° / 2 = 67.5°.
• If the minor arc AB is 225°, this is not possible as a minor arc. Minor arc must be < 180°.
• So, the minor arc AB is 135°.
• Therefore, ∠BAE = 135° / 2 = 67.5°.

Ответ:

• Угол BAE = 67.5°