In a circle where segments AC and BD are diameters, if the measure of arc AB is given as 3x – 70 degrees and the measure of arc DC is x + 10 degrees, how do we determine the measure of arc BC?

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To find the measure of arc BC in the scenario where circle O has diameters AC and BD, we first need to understand the relationships between the arcs and their associated angles.

We know the following:

  • Arc AB = 3x – 70 degrees
  • Arc DC = x + 10 degrees
  • Since AC and BD are diameters, arcs AB and DC are semicircles each measuring 180 degrees.

Since both of these arcs combined should equal the total degrees along the circumference of the circle, we can express the relationship mathematically:

Arc AB + Arc DC + Arc BC = 360 degrees

Since AC and BD are diameters, they divide the circle into four arcs: AB, BC, CD, and DA, where arcs AB and DC are already known. Thus, we can rewrite the equation as:

Arc AB + Arc DC + Arc BC = 360

Substituting the measures of arc AB and arc DC into the equation:

(3x – 70) + (x + 10) + Arc BC = 360

Combining like terms, we have:

4x – 60 + Arc BC = 360

Now, we isolate Arc BC:

Arc BC = 360 – (4x – 60)

Arc BC = 360 – 4x + 60

Arc BC = 420 – 4x

Now, to find the specific value of Arc BC, we need to calculate the value of x. We do this by using the information about the semicircles.

Since the total of arcs AB and BC equals 180 degrees (as they are semicircles), we can set up the equation:

(3x – 70) + (x + 10) = 180

Expanding and combining:

4x – 60 = 180

4x = 240

x = 60

Now we can substitute back to find Arc BC:

Arc BC = 420 – 4(60)

Arc BC = 420 – 240 = 180 degrees.

Therefore, the measure of arc BC is 180 degrees.

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