The vertices of triangle ABC lie on a circle with center O, angle ABC = 80 degrees, arc BC: arc AB = 3: 2.
| September 7, 2021 | education
The vertices of triangle ABC lie on a circle with center O, angle ABC = 80 degrees, arc BC: arc AB = 3: 2. Find the angles of triangle OAB.
The angles BOC and AOB are related as their arcs: that is, BOC: AOB = 1.5. Let’s take the angle AOB = x. Triangle AOB is isosceles because its sides are radii, and the angle ABO = BAO = (180 – x): 2.
The angle BOC = 1.5x, and since BOC is an isosceles triangle, then the angle CBO = (180 –1.5 x): 2. Then the angle ABO + CBO = (180 –1.5 x): 2 + (180 – x): 2 = 80. Hence x = 80.
Therefore, the angle AOB = 80 °, and the angle ABO = BAO = (180 – 80): 2 = 50 °.
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