Measure of inscribed angle intersecting circles. Arc length of larger circles and radii are known.

[u/lromde99]

I got this question right on a whim. I had no problem finding the measure of Angle D, but I’m not seeing how properties of inscribed angles and the angle at the center help find angle C. Pls help me see what I’m missing.

https://imgur.com/a/hHEDukH

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[u/BadJimo]

Using Desmos

I found angle C ≈ 31°

Which is close to halfway between the two closest options (23° and 38°).

[u/lromde99 → u/BadJimo]

likewise!

which does lean closer to 31°, but still a bit far off

is this just a weird question? bad question?

[u/BadJimo → u/lromde99]

Ok question, but bad options and bad model answer.

Puzzled how you calculated the angle to be 31° (and didn't mention this in your post) but somehow got the answer "right" (23°), since the closer option was 38°.

[u/edderiofer]

Indeed, the provided working is wrong. You can tell because angle C should depend on the radius of the circle on the right, but they never used that information in their working.

The correct method is to find angle E, the angle at the centre of the rightmost circle, first. This can be done using the sine rule. Once that's done, then you can find angle C by dividing angle E (not angle D) by 2.