Question

Отрезок BD - диаметр окружности с центром O . Хорда AC делит пополам радиус OB и перпендикулярна к нему . Найдите углы четырехугольника ABCD и градусные меры дуг AB , BC .

Answer 1

На чертеже, будет видно: что АВСО - ромб у которого все стороны равны радиусу, треугольники АВО и ВСО - равносторонние и углы которых равны 60 градусам. 
треугольник АСD - также равносторонний, он вписан в окружность и делит ее длину на три части, поэтому градусная мера дуг АD=СD=120 градусам. АВ=ВС=60 градусам.
проверка: 60+60+120+120=360 градусов

Углы 4-х угольника АВСD равны:
угол В = 60+60=120 градусам,
угол А = углу С = 30+60= 90 градусам. 
проверка : А+В+С+D= 90+120+90+60=360

Answer 2

Solution:

The problem describes a circle with center O, a chord AC that bisects the radius OB, and is perpendicular to it. We are asked to find the angles of the quadrilateral ABCD and the degree measures of the arcs AB and BC.

Let's start by analyzing the given information and solving the problem step by step.

1. Finding the Angles of Quadrilateral ABCD: - Since AC is a chord that bisects the radius OB and is perpendicular to it, we can conclude that triangle OAC is an isosceles right-angled triangle. - In an isosceles right-angled triangle, the angles are 45 degrees, 45 degrees, and 90 degrees. - Therefore, angle AOC = 90 degrees, angle OAC = 45 degrees, and angle OCA = 45 degrees.

2. Finding the Degree Measures of Arcs AB and BC: - The degree measure of an arc is equal to the central angle that intercepts it. - In this case, arc AB is intercepted by angle AOB, and arc BC is intercepted by angle BOC. - Since angle AOB and angle BOC are both right angles (90 degrees), the degree measures of arcs AB and BC are also 90 degrees each.

In summary, the angles of quadrilateral ABCD are: - Angle AOC = 90 degrees - Angle OAC = 45 degrees - Angle OCA = 45 degrees

The degree measures of arcs AB and BC are both 90 degrees.

I hope this helps! If you have any further questions or need additional assistance, feel free to ask.