An isosceles trapezoid is described around the circle. The angle for a larger base is 60 degrees. What is the ratio of the lengths of the bases?
According to the properties of a trapezoid circumscribed around a circle, the sum of the lengths of its lateral sides is equal to the sum of the lengths of its bases.
AB + CD = BC + AD.
Let’s draw the height of the trapezoid ВН. In a right-angled triangle ABН, the leg AH lies opposite the angle 30 and is equal to half the length of the hypotenuse AB. AH = AB / 2.
Since the trapezoid is isosceles, then DK = AB / 2.
Base AD = AH + НK + DK = AB / 2 + BC + AB / 2 = AB + BC.
By the property of an isosceles trapezoid circumscribed around a circle, the lateral side of the trapezium is equal to the midline of the trapezoid.
AB = (BC + AD) / 2.
AD = ((BC + AD) / 2) + BC.
AD = (BC + AD + 2 * BC) / 2.
AD * 2 = 3 * BC + AD.
AD = 3 * BC.
ВС / АD = 1/3.
Answer: The ratio of the bases of the trapezoid is 1/3.
One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.