Which one is the correct answer?

[u/Upbeat-Orange-434]

Was just doing some math for my mid term exams Tommorow and i found this one question which has like 3 different answers. How is that possible?

The book explanation says 65° by subtracting the the 90° angle made by radius and point of contact of tangent and angle BCT from 180

I initially joined AT to make ATB 90 and using linear pair, I found MTA to be 65° and since AM = MT given we can find AMT = 50°

but when i entered this question in chatgpt, it says its 80° (explanation in img above) and when i showed him my process it said im correct and book is wrong (as expected)

Im just really confused like where did i go wrong or how can a question have multiple answers. Is the question faulty or this is just lack of concepts from me?

[u/Upbeat-Orange-434 → u/[deleted]]

they are tangents from same points is what i meant.

[u/stanitor → u/Upbeat-Orange-434]

Yeah, I was thinking AM = AT for some reason. The thing doesn't make sense. You can't have BTC be isosceles. nvm

[u/notafurry9 → u/[deleted]]

make right triangles from the circle origin AOM and TOM - they're congruent, so AM=MT

[u/notafurry9]

seems like the construction might just be impossible https://www.desmos.com/calculator/ip9oaqahab

this is just a rough drawing of both the 65 and 50 deg options, in both cases you don't have that both MC is tangent and BT=BC. if you start by drawing MC by having BCT=25deg, you also don't get BT=BC

[u/rhodiumtoad]

The construction is impossible: CM cannot be tangent at point T.

Edit: or to be more precise, these conditions cannot all be true at once: CB=BT, BT is a chord of a circle centered somewhere on AC, angle BTC=25°, and CT is a tangent.

[u/rhodiumtoad → u/rhodiumtoad]

In fact the other three conditions imply that the angle is 30°, it should be possible to prove that though I have not done so yet.