math destroyer 2012 test 3 problem 10

[getsome111]

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Hey all, ive been lurking for a few months, ive found that most of the questions I have about the destroyers have been answered here and that has been quite useful, however I cant seem to find a solution that makes sense for this problem.



Given triangle ABC inscribed within a circle whose diameter AC forms one of the sides of the triangle. If Arc BC on the circle subtends an angle of 40 degrees, find the measure of angle BCA within the triangle.


The diagram only repeats the information in the question, a triangle ABC inscribed within a circle showing that line AC crosses the center of the circle and B as some arbitrary point.

I was under the impression that an Angle that subtends an arc length would be the angle touching the opposite side of the triangle. I have assumed that Angle BCA is representative of the subtended arc by BC because it is on the opposite side of the triangle(and thus should = 40 degrees as stated in the question stem). However in the first sentence of the solution it says "Since arc BC subtends an angle of 40 degrees, angle BAC measures half that amount, 20 degrees." what the heck, apparently the answer is 70 degrees for angle BCA. but from the get go this answer has thrown me off.. any help would be appreciated

 

[J777]

In geometry, an inscribed angle is going to be half the value of its intercepted arc. So if arc BC subtends a 40 degree angle, angle BAC is half that (i.e. 20). You must look at the angle opposite arc BC (angle BAC, not angle BCA).

So, we know that angle BAC = 20 degrees. Also, since AC is a diameter, arc AC must equal 180 degrees. Using the same rules as above, angle ABC must equal 180/2 = 90 degrees.

Now that you have two of the three angle measures, you can easily solve for the third. All three angles in a triangle need to add up to 180:

90 + 20 + BCA = 180
BCA = 70

 

[zammyd]

"So, we know that angle BAC = 20 degrees. Also, since AC is a diameter, arc AC must equal 180 degrees. Using the same rules as above, angle ABC must equal 180/2 = 90 degrees."

I'm really having trouble understanding this, can anyone explain?

 

[100%permwinner]

"So, we know that angle BAC = 20 degrees. Also, since AC is a diameter, arc AC must equal 180 degrees. Using the same rules as above, angle ABC must equal 180/2 = 90 degrees."

I'm really having trouble understanding this, can anyone explain?

This is one of the rarely tested topics. I learned this way back in 10th grade. Hope this video may be helpful.
https://www.khanacademy.org/math/ge...-circumscribed/v/inscribed-and-central-angles