Geometry Proof

Let the side length of a regular octagon (an 8-sided polygon) be $a$ . Prove that the area of this regular octagon is $2a^2 (1+\sqrt2)$ .

#Geometry

Easy Math Editor

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

All the eight triangles are congruent isosceles triangles since it is a regular octagon.Since angle of each triangle(the angle opposite to the sides of the octagon).Now take one of those triangles and use cosine rule to find the two equal lengths of that triangle.let those sides be $y.$ So, $a^2=y^2+y^2-2\times y\times y\times cos 45\Rightarrow y^2=\dfrac{a^2\sqrt2}{2(\sqrt2-1)}.$
Now we use sin rule for the same triangle.So, $Area(A)=\dfrac{1}{2}\times y^2\times cos45=\dfrac{1}{2}\times \dfrac{a^2\sqrt2}{2(\sqrt2-1)}\times \dfrac{1}{\sqrt2}=\dfrac{a^2}{4(\sqrt2-1)}.$ Now since all the triangles are congruent,the area of the Octagon $=8\times \dfrac{a^2}{4(\sqrt2-1)}=\dfrac{2a^2}{\sqrt2-1}.$ Rationalizing the denominator we get it to be $\boxed {2a^2(1+\sqrt2)}.$ Hence Proved.

Ayush G Rai - 4 years, 7 months ago

Can you do it using pure geometry . Without using trigonometry . It is under the head geometry.

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

Well i dont have any idea about that.

Ayush G Rai - 4 years, 7 months ago

@Ayush G Rai – I can do that , but I want to know how post image file from my computer.

Ahmad Saad - 4 years, 7 months ago

@Ahmad Saad – you can read this one.https://brilliant.org/math-formatting-guide/#

Ayush G Rai - 4 years, 7 months ago

@Ayush G Rai – sorry, I read that but I still don't know how to use that. Thanks

Ahmad Saad - 4 years, 7 months ago

@Ahmad Saad – You must know how to post a solution to a general question as i have seen your solution . Just solve any easy question on brilliant you will have the right to post a solution and just insert a image there (the image of solution to this geometricc proof) you see that https...... something appears there copy that paste here in the comments and just post it ..................................................................

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – Thanks.

Ahmad Saad - 4 years, 7 months ago

@Ahmad Saad – Well , I have a solution regarding this question . But I am messed up in an another geometry question right now . Quadrilateral in a triangle and I am not able to point out the mistake . can you help me ?

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω –

Ahmad Saad - 4 years, 7 months ago

@Ahmad Saad – Ar. BQD = 154/3 is not correct it should be 77/3

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – You are also telling the same thing . I have posted a solution by clearing how to proceed in similar manner . It will become too long so i haven't repeated it again and again .

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – Now i have arrived at my mistake

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – well thanks.now even i understood some tricks to solve problems.That was a nice problem and also construction.Any more geo problems

Ayush G Rai - 4 years, 7 months ago

@Ayush G Rai – A similar question but somehow easier is A Classical geometry problem (level hard - geometry )

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Ayush G Rai – Well , I have a solution regarding this question . But I am messed up in an another geometry question right now . Quadrilateral in a triangle and I am not able to point out the mistake . can you help me ?

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – can u give me only the question??all the details and informations

Ayush G Rai - 4 years, 7 months ago

@Ayush G Rai – You can click on the link above Quadrilateral in a triangle above in my post . or you can search it as a level 5 problem .

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Ayush G Rai – Well , I also need to rectify my solution . Please it is not too long and I am unable to point out the mistake

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – can explain how u got ar.BPD,ar.BRE.... in the last statement.I am not able to figure the similar manner.

Ayush G Rai - 4 years, 7 months ago

@Ayush G Rai – Well , just in the similar manner as i have drawn ST || AC just like that drawing PM || AC and using the same technique , then RN ||\ AC and so on

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – point M is to the left of E or right?

Ayush G Rai - 4 years, 7 months ago

@Ayush G Rai – It is to the right .

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – But that does not matters , if you take it on left then DM = 8x/7 If you take it on right then EM = x/7 Both are same..............

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – I think so it is to the right of E

Ayush G Rai - 4 years, 7 months ago

@Ayush G Rai – yeah

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Ayush G Rai – I was able to point out my mistake ,wasted 7 hours . I was doing a silly mistake Ar. BQD sholud be 77/3 and I was taking it to be 154 / 3

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – Yes u are right!

Ayush G Rai - 4 years, 7 months ago

Ahmad Saad - 4 years, 7 months ago

Even i wanted to show the same thing.

Ayush G Rai - 4 years, 7 months ago

nice

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

Lets try this problem https://brilliant.org/discussions/thread/rmo-2016/

Ayush G Rai - 4 years, 7 months ago

Sorry , I saw your question yesterday . I am trying it .

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

Please have a look at RMO question.

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

Does it ask to prove that BKCL are concyclic , if angle BAC = 90 degreess or vice versa .

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

It asks to prove that BKCL are concyclic , if angle BAC = 90 degrees

Ayush G Rai - 4 years, 7 months ago

@Ayush G Rai –

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – how do u know that KD=AD?

Ayush G Rai - 4 years, 7 months ago

@Vishwash Kumar Γξω – Good one...+1

Ayush G Rai - 4 years, 7 months ago

@Ayush G Rai – Thanks !!!

vishwash kumar - 4 years, 7 months ago

@Ayush G Rai – Well , i Forgot to mention something . △ BDK ~ △ LDC By SAS similarity criteria ( Not used very often )

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Ayush G Rai – Well I forgot to mention something

△ BDK ~ △ LDC ( By SAS similarity ) (Not used very often )

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Ayush G Rai – Have you applied for RMO (Sorry , I had earlier typed IMO by mistake )

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – yeah.I wrote it this year.Could solve only 2.[both geometry]What about u?

Ayush G Rai - 4 years, 7 months ago

@Ayush G Rai – I have came to know about this just now when I solved this queation I have had never heard about it earlier .

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Ayush G Rai – Do you know , what is the reflection of a point about a line ???????

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – yup.that is the easiest in co-geometry

Ayush G Rai - 4 years, 7 months ago

@Ayush G Rai – Yeah ,I do know that . You mean the image of a point in the cartesian plane .

vishwash kumar - 4 years, 7 months ago

@Vishwash Kumar – But I am asking you about the reflection of any point about a line on euclidean plane

vishwash kumar - 4 years, 7 months ago

@Vishwash Kumar – I was solving a question " What is this line ? " On Brilliant when I encountered this term .

vishwash kumar - 4 years, 7 months ago

@Vishwash Kumar – I took it easy and assumed that when a perpendicular drawn from the (actual) point to line is produced and a point at the same distance from the line (initial) as the actual point was from the line is cut off is the reflection of the point about that line .

vishwash kumar - 4 years, 7 months ago

@Vishwash Kumar – I think that it was my misconception

vishwash kumar - 4 years, 7 months ago

@Vishwash Kumar – So asked you .

vishwash kumar - 4 years, 7 months ago

@Ayush G Rai – Okay , then I have solved

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – Well even i solved it but iam lazy to write the solution

Ayush G Rai - 4 years, 7 months ago

@Ayush G Rai – Same here .

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Ayush G Rai – But I am gonna post it .

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Ayush G Rai – How did you do it ??

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – We note that the segments BC and KL cannot coincide, because the same would imply that AD is a perpendicular bisector of seg BC, and that ∆ABC is isosceles, contradicting the given information that the triangle is scalene. So without loss of generality, let A − B − K and A − L − C. We will separately prove the two parts of this problem:

Ayush G Rai - 4 years, 7 months ago

@Ayush G Rai – Yeah , I was able to write my solution in 1 1/2 hours .

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Ayush G Rai – Hats OFF to my writing speed .

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Ayush G Rai – I have a different solution .

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Ayush G Rai – Remember that in any right triangl the length of the line segment joining the midpoint of the hypo. to opposite vertex (containing right angle ) = 1/2 * hypotenuse

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – Remember that in any right triangl the length of the line segment joining the midpoint of the hypo. to opposite vertex (containing right angle ) = 1/2 * hypotenuse

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Ayush Rai I deactivated my account and forgot to ask a very important thing from you people. So, I broke my computer's firewall and opened Brilliant (hahaha). I had to ask you: Who is the regional coordinator of RMO? Is it someone like vice chancellor of the university or anyone else? I searched on internet but got nothing. I think it would be the person with the same post for every region. And one another thing is there any deadline to fill RMO form? I googled it and got that the filling started on 15th January. I was releaved thinking that the RMO is held in december so I have much time but no.

Vishwash Kumar ΓΞΩ - 4 years, 4 months ago

the RMO coordinators are different for different regions.U can check this out

Ayush G Rai - 4 years, 4 months ago

@Ayush G Rai – Hey you have given me the link to MP team.

Vishwash Kumar ΓΞΩ - 4 years, 4 months ago

@Vishwash Kumar Γξω – oh..really .idk

Ayush G Rai - 4 years, 4 months ago

@Ayush G Rai – Are you kidding me?

Vishwash Kumar ΓΞΩ - 4 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$