To solve for the measure of arc BC in circle O, we start by analyzing the information given. We know that line segments AC and BD are diameters of the circle. Therefore, arcs AB and DC are related to these diameters.
From the problem statement, we have:
- Measure of arc AB = 3x – 70 degrees
- Measure of arc DC = x + 10 degrees
Since AC and BD are diameters, they divide the circle into two equal halves. The sum of the measures of arc AB and arc DC should equal 180 degrees because they represent one half of the circle. Therefore, we can set up the following equation:
(Measure of arc AB) + (Measure of arc DC) = 180 degrees
Substituting the expressions we have:
(3x – 70) + (x + 10) = 180
Simplifying this equation:
- 3x – 70 + x + 10 = 180
- 4x – 60 = 180
- 4x = 240
- x = 60
Now, substituting x back into the expressions for arcs AB and DC:
Measure of arc AB = 3(60) – 70 = 180 – 70 = 110 degrees
Measure of arc DC = 60 + 10 = 70 degrees
To find the measure of arc BC, we note that arc BC is the remaining arc in the semicircle:
Measure of arc BC = 180 degrees – (Measure of arc AB + Measure of arc DC)
Substituting the values we calculated:
Measure of arc BC = 180 – (110 + 70) = 180 – 180 = 0 degrees
Therefore, the measure of arc BC is 0 degrees, indicating that points B and C are the same point in this case, which aligns with the definitions of diameters in the circle.