Geometry Question from exam 2

[fbmagnumopus]

Hey Tommy,

I wanted to ask about this question from my second practice test, question 9 from the 2nd math section:

The length of arc AXB is twice the length of arc BZC, and the length of arc AYC is three times the length of arc AXB. What is the measure of angle BCA?

(9 Lengths of Arcs Devilish 3:06 13:23 DQ Circles & Cylinders )

Why do we take the sum of the angle measures of the arcs to be equal to 180? The angle measure of the arc is equal to the angle measure of its central angle. So when we add up all the angle measures they should equate to 360 not 180 right?

This would lead to a final answer of 80 rather than 40.

[tommywallach]

Hey FB,

This is because a triangle has 180 degrees in it--nothing more complex than that.

HOWEVER, I do know what has you confused. What you're thinking of is if we made angles from the CENTER of the circle. Try it right now. Draw this figure as it is in the question. Now, put a point in the center of the circle, and draw lines to the triangle. NOW, if you were to add up all THOSE angles, it would be 360. It turns out, this is a rule of circles. Pay close attention, because it's a little wacky to write out:

The central angle of a circle is twice any inscribed angle subtended by the same arc.

Basically, the angle made by point C on the triangle would be DOUBLE if it only went to the center (instead of all the way out to the circumference of the circle).

Hope that helps!

-t

[fbmagnumopus]

Hey Tommy,

Yes you're right . I think thats what confused me too.

And it's kinda confusing which approach one should use here , start with the triangle or start by considering the arc-central angle- intercepted angle relation. I tried the latter Again this time and I got 40 as the right answer. So thanks for clarifying that.

I think some people like me would have trouble recognizing that the relationships between the arcs apply for the intercepted angles because the extra factor of 2 coming in due to the intermediary central angle connecting them but since its effect cancels out in the end for this question, it shouldn't affect the answer .

[tommywallach]

Hey FB,

It's definitely a confusing one. In this case, the triangle method is much friendlier; we wouldn't even need that crazy complex rule!

-t