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Accept answers at your own risk. We use automoderator for a few things. It may delete a post erroneously. If your post is missing and you're sure it's not a rule-breaker, send us a note and we'll look into the issue. We ban all bots. Embarrassed engineer dad here, trying to help daughter with some algebra. Ran into this problem which, to me, appears to have incomplete information.
She was trying to solve it assuming that EF and ED are the same length and therefore everything can just be split in half and solved easily. But I redrew it to show that she could have the same angles given but have a very lopsided drawing as well. I have a feeling we want to extend ED and EF out to that tangent line, but I tried that and just end up with more unknown angles.
I'm going to take a stab at it. See if you can follow my logic, and tell me if this is sound to you: Imagine a line CE.
Of course if this figure isn't symmetrical, then there isn't enough information to solve this. Since this is high school geometry I think it's safe to assume that it is. If you could prove that, then the bisecting strategy would work.
This is an interesting question, I feel like there is a possible solution with the given info but it's certainly not obvious. I tried extending the line segments CD and CF out to the tangent line, but as you said, it just gives you more unknown angles.
Sorry man, I have no idea. The problem with this is you need to prove it's symmetry. It's been a while since I have done formal geometry, so I don't remember if this is something that can be assumed. Yup, that's definitely an issue. If it wasn't symmetrical, then there is definitely not enough information to solve this. But because it's high school geometry, I think it's a safe assumption that it is.
I doubt they would be asking a high schooler to prove symmetry in this case. Yes, if he could prove that the angles DCA and ACF were equal to each other, then the bisecting strategy would definitely work.
I'm thinking about how to do that but can't figure it out. That's the problem, they don't have to be symmetrical I'm assuming C is the center of the circumference here. What are some of the theorems they are currently working with.
I don't want to be trying to use arcs and inscribed angles if they haven't gotten there yet. The rest of the assignment uses only: I found this page which does seem to somehow prove that if the angle at the center is double the angle where the chords meet, then the chords must be equal length:.
But they start with 2 tangents that meet at a given angle and work back from there. I can't quite make the connection to my problem though. And that example seems to have more information to start. I think this will be your jumping off point. I'm going to start on it now, but I figured I would share the info as I found it. I'm assuming that ED and EF are equal in magnitude as you described already.
Also, I'm focusing on the triangle CEF. Therefore, using the fact that a straight line is degrees, angle ECF is degrees. Well since this isn't high-level math, there is no reason we can't assume what is obvious in the design of the question.
Since there is no other way to do it without more information, we can assume it is the only way. The entire point of formal geometry a sophomore level class, often taken by freshman is to being with the question and the answer and prove the answer. Using your "since this isn't high-level math" argument we could just say bust out a protractor and measure the angle. What you are missing is the entire point of the question, which is to prove that the figure is symmetrical first.
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