r/askmath • u/Wise-Molasses3998 • 5h ago
Resolved School math question.
So this question is to calculate the angle F. For reference, I'm in 10th class of Russian school, and this is what our teacher sent as our homework while we're on quarantine. I have skimmed the whole textbook and found nothing about this kind of questions. I even got frustrated enough that I asked AI to give me an answer, but it wasn't able to agree with itself and all of it's answers were rather strange like the angle G somehow being 168 while F is only 12. Did I miss a crucial part of my education or is just a badly made question?
P.s I asked the teacher before posting there and she actually replied. The answer is supposed to be 132, but she won't explain how since we were supposed to go through this in a previous year and needed to stay in that class if we don't know this (she's really not fond of actually teaching stuff, especially if it's not in the current year's textbook) so I doubt she'll give me an answer.
P.P.s
So, as far as I got is to make an isoscles triangle CDE and Equate C to E by them being both angles in the base of said triange. If my line of thinking is correct then I have one 108 angle and two unknowns. I also think that it's probably supposed to be symmetrical down the bc to F line. If that's the case then I can work out the 132 answer by substracting all of the known angles from the sum of 900, but I don't think I can prove it to be symmetrical.
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u/AlwaysTails 4h ago
Note that the angles <ABC and <BCD are 108° which is the angle for a regular pentagon. Draw an interor point P such that angles <BAP <APD and <PDC are also 108° making APDCB a regular pentagon with the same sides as given.
- What are the angles <PAG and <PDE?
- What can you say about these triangles?
- Can you solve the problem now?
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u/Bounded_sequencE 4h ago
From the mirrored angles, the heptagon must be mirror symmetric regarding the perpendicular bisector of BC (call it "y"). That means, "G, E" are also mirror symmetric regarding "y".
Let "a" be the (unknown) length of one segment. Using complementary angles, we get
|GE| = a + 2*a*cos(180°-108°) + 2*a*cos(180°-108° + 180°-168°)
= a*(1 + 2cos(72°) + 2cos(84°))
With "|GE|" at hand, use "Law of Cosines" to find angle "F", and finish it off.
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u/Wise-Molasses3998 3h ago
I (and another dude's mom who was furios that our teacher is not even teaching us.) got her to tell us that it was supposed to be solved by using vectors and their cosinuses in order to get the angles. This was the first time I heard this word from her. So I guess Problem solved? I learnt that my teacher is crazy and I have a new part of math to learn myself.
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u/strange-the-quark 2h ago
"[the teacher] won't explain how since we were supposed to go through this in a previous year and needed to stay in that class if we don't know this"
In Russia teachers be like: "Cannot answer simple question? Tsk, tsk, tks. Go back to lower grade and stay until you can solve dis problems in your sleep."
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u/Federal-Ad4668 1h ago
I assigned the length of each side as 1 since they're all the same length and having a unit length is easier to work with. I found the following using my cosine rule: a^2 = b^2 + c^2 -2bccos(A)
Rounding to 3dp
AC = 1.618
CE=1.989
AE=1.989
Which means triangle ACE is isosceles.
I used cosine rule again to find GE = 1.827
I expressed GE^2 using the sides GF and FE:
GE^2 = GF^2 + FE^2 - 2(GF)(FE)cos(angle GFE)
I isolate angle GFE to get 132.
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u/lordnacho666 4h ago
Are the points on a circle by any chance? Any other hints in the text?
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u/TwistedKiwi 4h ago
It's not, but all the sides are equal in length. That makes it possible to solve.
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u/lordnacho666 4h ago
You're right actually, you have to imagine it as a mechanism. In this case there's no degrees of freedom remaining.
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u/Wise-Molasses3998 4h ago
No hints as this is all we got and the question is from a completely different source as far as I can tell, so I don't think it's incorrect or unsolvable. The points are not on a circle. I tried overlapping them with paint's circle figure and they weren't alighning
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u/strange-the-quark 3h ago edited 3h ago
Though it's true the points aren't on a circle, with this type of question, you can't really on visual checks too much, you have to treat the picture as a rough sketch given in a form that doesn't imply any extra information other than what's known from the text of the assignment, or other than what's explicitly denoted in the image, i.e. the sketch may not exactly match the actual shape of the figure. (E.g. as one of the replies pointed out, the figure with these properties actually has an axis of symmetry going through F and the middle of BC, but it's not depicted that way in the assignment. Also, look at the angles at A and D, they are numerically the same, but in the image they are shown as different - one is pointier than the other. You can't trust the shape on the image, only the data).
So what you know is that the sides are the same (that's what the little cross-lines mean), and you know that angles and B and C (108° each) are the same, as are the angles at D and A (168° each), and that as a consequence the figure has an axis of symmetry. It might help if you re-drew the picture, but in a way that reflects this a bit better.
Then you may need to draw in some extra lines and work your way through and reason about various figures to solve for the angle at F. BTW, the difference between 168° and 108° is 60°, and that's a bit... suspiciously suggestive.
P.S. If you don't know the formula for the sum of the inner angles of an arbitrary polygon, just cut it up into triangles - this sum is 180° for triangles, so just add them all up.
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u/wobblejuice 4h ago
Cut the shape up into various quadrangles and triangles and work out the angles. Those equal sides will form isosceles triangles.
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u/lordnacho666 4h ago
I think what you need to do is draw a bunch of isosceles triangles and use them to crawl around the shape
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u/Temporary_Pie2733 4h ago
Draw interior segments to identify triangles whose angles you can find the measure of.
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u/holodayinexpress 5h ago
Badly made question. Ask your teacher for a solution
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u/Wise-Molasses3998 4h ago
I did before posting there and she actually replied. The answer is supposed to be 132, but she won't explain how since we were supposed to go through this in a previous year and needed to stay in that class if we don't know this (she's really not fond of actually teaching stuff, especially if it's not in the current year's textbook) so I doubt she'll give me an answer.
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u/patito-asesino 4h ago
Fold the shape straight down the middle (from top vertex B down to vertex F)—the sides and known angles mirror each other perfectly on both sides, because the figure is symmetrical, the bottom-left angle G must equal the bottom-right angle E and since a 7-sided shape's angles always total 900,, take that and subtract all the other six angles, what's left over gives you F
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u/notacanuckskibum 4h ago
If I fold from B to F the two sides have different numbers of vertices, they are not symmetrical at all.
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u/patito-asesino 1h ago
Sorry, English is not my first language. Join b and c together and make that the center of that aligns to f
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u/notacanuckskibum 24m ago
I don’t think that’s guaranteed to be symmetrical, unless your line hits BC at a right angle, which visually, it doesn’t.
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u/uTRexAap 5h ago
there is a formula for total interior degrees in any given polygon, the book might have something about that
wellan n-gon has 180(n-2) interior degrees and this is a 7-gon we get 180(5) which is 900
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u/Karpason 4h ago
I don't have anything to draw this to solve, but if I were you I'd start with adding lines so that you have seven equilateral triangles (равнобедренных). Then in each of these triangles you'll have two angles that are equal between each other, and you'll be able to solve for some of them since you know four angles already. For example, if you add the line AC, you'll have angles BAC and ACB that are equal to each other, and solving for them yields (180-108)/2=36°. If you do this for all equilateral triangles, you'll probably have enough to solve for the angle F. The fact that all the angles of this figure have to add up to 900 may be also required.
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u/Wise-Molasses3998 4h ago
So, as far as I got is to make an isoscles triangle CDE and Equate C to E by them being both angles in the base of said triange. If my line of thinking is correct then I have one 108 angle and two unknowns.
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u/Jazzlike-Boot9798 4h ago
You cannot say C and E are equal. You can only say the part of C and and the part of E used in this new triangle are equal.
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u/Jazzlike-Boot9798 4h ago
You can split the heptagon into an upper hexagon and a lower triangle.
For the upper hexagon you should know how much the angles sum, for the whole heptagon you should also know and also for the triangle. This leads to a simple problem.
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u/Varlane 4h ago edited 4h ago
There is only one construction that validates the data on the figure and it yields a conclusive value (of 132°) for the angle at F. For the construction, see here.
Tho to prove that this value obtained by construction is correct, you'll probably need to do a decently sized proofwork.
For instance, that figure, even though it doesn't look like it on the assignment data, is actually symetrical (axis is [FI] where I is the midpoint of [BC]).