Well, this problem seems like a real angle grinder! Let's see if we can find the measure of the third arc together.
First, let's call the measure of the larger arc intercepted by the secant and tangent as 7x, and the measure of the smaller arc as 3x.
Since the secant and tangent intersect in a 42-degree angle, we can use one of the pretzel-like properties of a circle to find the measure of each of the intercepted arcs.
The angle formed by the secant and its intercepted arc is half the measure of the intercepted arc. So, the angle formed by the smaller arc is 3x/2 degrees. Similarly, the angle formed by the larger arc is 7x/2 degrees.
Now it's time to unleash some math skills! Since the secant and tangent intersect in a 42-degree angle, we know that the angles formed by the secant (3x/2) and tangent (7x/2) must add up to 42.
So, we can write an equation: (3x/2) + (7x/2) = 42.
To simplify, we can combine like terms: (10x/2) = 42.
Dividing both sides by 5, we find that x = 8.4.
Now we can determine the measure of the third arc. Since the three arcs add up to 360 degrees in a circle, we have:
3x + 7x + Third Arc = 360.
Simplifying, we get 10x + Third Arc = 360.
Substituting the value of x we found earlier, we have 10 * 8.4 + Third Arc = 360.
So, 84 + Third Arc = 360.
Finally, we can subtract 84 from both sides to find the measure of the third arc:
Third Arc = 360 - 84.
Calculating that, we have the answer: The measure of the third arc is 276 degrees.
I hope this explanation brought some joy to your mathematical journey!