Final answer:
The measure of arc AE is 180 degrees, which is found by calculating the measure of arcs AC and BC and then using the total to subtract from the full circle's measure of 360 degrees. There was a discrepancy in the question's options as none of them match the calculated answer.
Step-by-step explanation:
The measure of arc BC is given as (0.5x + 34) degrees, and since AD and BE are diameters in circle O, the measure of arc AE is the measure of a semicircle, which is 180 degrees. Given that angle AOC is a right angle, arc AC corresponds to 90 degrees because a radius that rotates through a right angle sweeps out a quarter of the circle's circumference. The measure of arc ABC is therefore the sum of the measures of arcs AC and BC. Since arc AC is a 90-degree arc (a quarter circle), we can write the measure of arc ABC as 90 + (0.5x + 34). The sum of the measures of arc ABC and arc AE must equal 360 degrees because together they cover the entire circle. Thus, we calculate the measure of arc ABC first and then use it to find the measure of arc AE.
We set up the equation: 180 + 90 + (0.5x + 34) = 360.
270 + 0.5x + 34 = 360
0.5x + 304 = 360
0.5x = 56
x = 112.
Since the measure of arc BC is (0.5x + 34) degrees, we substitute x with 112 to find measure of arc BC.
Measure of arc BC = (0.5 * 112) + 34 = 56 + 34 = 90 degrees.
Now, we can find the measure of arc AE as follows:
360 - (measure of arc ABC) = measure of arc AE
360 - (90 + 90) = measure of arc AE
measure of arc AE = 360 - 180 = 180 degrees.
Hence, the correct answer to the question 'What is mArc AE?' is 180 degrees, which is not one of the options provided; thus, either the question has a typo regarding the options, or there's a misunderstanding in the given information.