In regular polygon ABCDE…, we have ∠ACD = 120°. How many sides does the polygon have?

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In regular polygon ABCDE..., we have ∠ACD = 120°. How many sides does the polygon have?

How would I start with solving this? AC would have different lengths depending on the polygon.

geometry polygons

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asked Jul 20 at 6:19

ShadyAF

288

add a comment | 

up vote
0
down vote

favorite

In regular polygon ABCDE..., we have ∠ACD = 120°. How many sides does the polygon have?

How would I start with solving this? AC would have different lengths depending on the polygon.

geometry polygons

share|cite|improve this question

asked Jul 20 at 6:19

ShadyAF

288

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

In regular polygon ABCDE..., we have ∠ACD = 120°. How many sides does the polygon have?

How would I start with solving this? AC would have different lengths depending on the polygon.

geometry polygons

share|cite|improve this question

asked Jul 20 at 6:19

ShadyAF

288

In regular polygon ABCDE..., we have ∠ACD = 120°. How many sides does the polygon have?

How would I start with solving this? AC would have different lengths depending on the polygon.

geometry polygons

share|cite|improve this question

asked Jul 20 at 6:19

ShadyAF

288

share|cite|improve this question

share|cite|improve this question

asked Jul 20 at 6:19

ShadyAF

288

asked Jul 20 at 6:19

ShadyAF

288

asked Jul 20 at 6:19

ShadyAF

288

288

add a comment | 

add a comment | 

2 Answers
2

active

oldest

votes

up vote
1
down vote



accepted

Assume that there exist totally $n$ sides along the vertexes $D,E,cdots,A$. Notice that $$120^o=angle ACD oversetm=frac12widehatDEcdots A=frac12cdot 360^ocdot fracnn+3.$$

Thus, $n=6.$ As a result, the number of the sides of the polygon is $n+3=9.$

share|cite|improve this answer

edited Jul 20 at 6:42

answered Jul 20 at 6:29

mengdie1982

2,912216

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Alright what the hell. I was adding a comment and it edited your answer instead... I'm sorry if that deleted any changes you've made, it was not my intention. EDIT: I see now, it popped up an 'edit summary box' right where my comment box was when I went to copy/paste some LaTeX and I happily added my 'comment' there.
  – orlp
  Jul 20 at 6:33
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Either way, I'm not too familiar with geometric notation, could you explain what $oversetm =$ and $widehatDEcdots A$ mean?
  – orlp
  Jul 20 at 6:35
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yep. That is to say, $angle ACD$ equals half of the central angle opposite to the arc $DEcdots A$.
  – mengdie1982
  Jul 20 at 6:40
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  did you mean n=9?
  – ShadyAF
  Jul 20 at 6:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ShadyAF yes,a typo.
  – mengdie1982
  Jul 20 at 6:43
  

  
  
  
  

add a comment | 

up vote
0
down vote

The interior angle of a regular polygon is $$angle ABC=angle BCD=frac180(n-2)n$$
Now the triangle ABC is isosceles, so $$angle BCA = frac12(180-frac180(n-2)n) \ angle BCA+angle ACD = angle BCD \ frac12(180-frac180(n-2)n)+120=frac180(n-2)n$$
This gives $n=9$.

share|cite|improve this answer

answered Jul 20 at 6:39

Piyush Divyanakar

3,258122

add a comment | 

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote



accepted

Assume that there exist totally $n$ sides along the vertexes $D,E,cdots,A$. Notice that $$120^o=angle ACD oversetm=frac12widehatDEcdots A=frac12cdot 360^ocdot fracnn+3.$$

Thus, $n=6.$ As a result, the number of the sides of the polygon is $n+3=9.$

share|cite|improve this answer

edited Jul 20 at 6:42

answered Jul 20 at 6:29

mengdie1982

2,912216

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Alright what the hell. I was adding a comment and it edited your answer instead... I'm sorry if that deleted any changes you've made, it was not my intention. EDIT: I see now, it popped up an 'edit summary box' right where my comment box was when I went to copy/paste some LaTeX and I happily added my 'comment' there.
  – orlp
  Jul 20 at 6:33
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Either way, I'm not too familiar with geometric notation, could you explain what $oversetm =$ and $widehatDEcdots A$ mean?
  – orlp
  Jul 20 at 6:35
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yep. That is to say, $angle ACD$ equals half of the central angle opposite to the arc $DEcdots A$.
  – mengdie1982
  Jul 20 at 6:40
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  did you mean n=9?
  – ShadyAF
  Jul 20 at 6:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ShadyAF yes,a typo.
  – mengdie1982
  Jul 20 at 6:43
  

  
  
  
  

add a comment | 

up vote
1
down vote



accepted

Assume that there exist totally $n$ sides along the vertexes $D,E,cdots,A$. Notice that $$120^o=angle ACD oversetm=frac12widehatDEcdots A=frac12cdot 360^ocdot fracnn+3.$$

Thus, $n=6.$ As a result, the number of the sides of the polygon is $n+3=9.$

share|cite|improve this answer

edited Jul 20 at 6:42

answered Jul 20 at 6:29

mengdie1982

2,912216

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Alright what the hell. I was adding a comment and it edited your answer instead... I'm sorry if that deleted any changes you've made, it was not my intention. EDIT: I see now, it popped up an 'edit summary box' right where my comment box was when I went to copy/paste some LaTeX and I happily added my 'comment' there.
  – orlp
  Jul 20 at 6:33
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Either way, I'm not too familiar with geometric notation, could you explain what $oversetm =$ and $widehatDEcdots A$ mean?
  – orlp
  Jul 20 at 6:35
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yep. That is to say, $angle ACD$ equals half of the central angle opposite to the arc $DEcdots A$.
  – mengdie1982
  Jul 20 at 6:40
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  did you mean n=9?
  – ShadyAF
  Jul 20 at 6:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ShadyAF yes,a typo.
  – mengdie1982
  Jul 20 at 6:43
  

  
  
  
  

add a comment | 

up vote
1
down vote



accepted

up vote
1
down vote



accepted

Assume that there exist totally $n$ sides along the vertexes $D,E,cdots,A$. Notice that $$120^o=angle ACD oversetm=frac12widehatDEcdots A=frac12cdot 360^ocdot fracnn+3.$$

Thus, $n=6.$ As a result, the number of the sides of the polygon is $n+3=9.$

share|cite|improve this answer

edited Jul 20 at 6:42

answered Jul 20 at 6:29

mengdie1982

2,912216

Assume that there exist totally $n$ sides along the vertexes $D,E,cdots,A$. Notice that $$120^o=angle ACD oversetm=frac12widehatDEcdots A=frac12cdot 360^ocdot fracnn+3.$$

Thus, $n=6.$ As a result, the number of the sides of the polygon is $n+3=9.$

share|cite|improve this answer

edited Jul 20 at 6:42

answered Jul 20 at 6:29

mengdie1982

2,912216

share|cite|improve this answer

share|cite|improve this answer

edited Jul 20 at 6:42

edited Jul 20 at 6:42

edited Jul 20 at 6:42

answered Jul 20 at 6:29

mengdie1982

2,912216

answered Jul 20 at 6:29

mengdie1982

2,912216

answered Jul 20 at 6:29

mengdie1982

2,912216

2,912216

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Alright what the hell. I was adding a comment and it edited your answer instead... I'm sorry if that deleted any changes you've made, it was not my intention. EDIT: I see now, it popped up an 'edit summary box' right where my comment box was when I went to copy/paste some LaTeX and I happily added my 'comment' there.
  – orlp
  Jul 20 at 6:33
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Either way, I'm not too familiar with geometric notation, could you explain what $oversetm =$ and $widehatDEcdots A$ mean?
  – orlp
  Jul 20 at 6:35
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yep. That is to say, $angle ACD$ equals half of the central angle opposite to the arc $DEcdots A$.
  – mengdie1982
  Jul 20 at 6:40
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  did you mean n=9?
  – ShadyAF
  Jul 20 at 6:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ShadyAF yes,a typo.
  – mengdie1982
  Jul 20 at 6:43
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Alright what the hell. I was adding a comment and it edited your answer instead... I'm sorry if that deleted any changes you've made, it was not my intention. EDIT: I see now, it popped up an 'edit summary box' right where my comment box was when I went to copy/paste some LaTeX and I happily added my 'comment' there.
  – orlp
  Jul 20 at 6:33
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Either way, I'm not too familiar with geometric notation, could you explain what $oversetm =$ and $widehatDEcdots A$ mean?
  – orlp
  Jul 20 at 6:35
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yep. That is to say, $angle ACD$ equals half of the central angle opposite to the arc $DEcdots A$.
  – mengdie1982
  Jul 20 at 6:40
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  did you mean n=9?
  – ShadyAF
  Jul 20 at 6:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ShadyAF yes,a typo.
  – mengdie1982
  Jul 20 at 6:43
  

  
  
  
  

Alright what the hell. I was adding a comment and it edited your answer instead... I'm sorry if that deleted any changes you've made, it was not my intention. EDIT: I see now, it popped up an 'edit summary box' right where my comment box was when I went to copy/paste some LaTeX and I happily added my 'comment' there.
– orlp
Jul 20 at 6:33

Alright what the hell. I was adding a comment and it edited your answer instead... I'm sorry if that deleted any changes you've made, it was not my intention. EDIT: I see now, it popped up an 'edit summary box' right where my comment box was when I went to copy/paste some LaTeX and I happily added my 'comment' there.
– orlp
Jul 20 at 6:33

Either way, I'm not too familiar with geometric notation, could you explain what $oversetm =$ and $widehatDEcdots A$ mean?
– orlp
Jul 20 at 6:35

Either way, I'm not too familiar with geometric notation, could you explain what $oversetm =$ and $widehatDEcdots A$ mean?
– orlp
Jul 20 at 6:35

Yep. That is to say, $angle ACD$ equals half of the central angle opposite to the arc $DEcdots A$.
– mengdie1982
Jul 20 at 6:40

Yep. That is to say, $angle ACD$ equals half of the central angle opposite to the arc $DEcdots A$.
– mengdie1982
Jul 20 at 6:40

did you mean n=9?
– ShadyAF
Jul 20 at 6:42

did you mean n=9?
– ShadyAF
Jul 20 at 6:42

@ShadyAF yes,a typo.
– mengdie1982
Jul 20 at 6:43

@ShadyAF yes,a typo.
– mengdie1982
Jul 20 at 6:43

add a comment | 

up vote
0
down vote

The interior angle of a regular polygon is $$angle ABC=angle BCD=frac180(n-2)n$$
Now the triangle ABC is isosceles, so $$angle BCA = frac12(180-frac180(n-2)n) \ angle BCA+angle ACD = angle BCD \ frac12(180-frac180(n-2)n)+120=frac180(n-2)n$$
This gives $n=9$.

share|cite|improve this answer

answered Jul 20 at 6:39

Piyush Divyanakar

3,258122

add a comment | 

up vote
0
down vote

The interior angle of a regular polygon is $$angle ABC=angle BCD=frac180(n-2)n$$
Now the triangle ABC is isosceles, so $$angle BCA = frac12(180-frac180(n-2)n) \ angle BCA+angle ACD = angle BCD \ frac12(180-frac180(n-2)n)+120=frac180(n-2)n$$
This gives $n=9$.

share|cite|improve this answer

answered Jul 20 at 6:39

Piyush Divyanakar

3,258122

add a comment | 

up vote
0
down vote

up vote
0
down vote

The interior angle of a regular polygon is $$angle ABC=angle BCD=frac180(n-2)n$$
Now the triangle ABC is isosceles, so $$angle BCA = frac12(180-frac180(n-2)n) \ angle BCA+angle ACD = angle BCD \ frac12(180-frac180(n-2)n)+120=frac180(n-2)n$$
This gives $n=9$.

share|cite|improve this answer

answered Jul 20 at 6:39

Piyush Divyanakar

3,258122

The interior angle of a regular polygon is $$angle ABC=angle BCD=frac180(n-2)n$$
Now the triangle ABC is isosceles, so $$angle BCA = frac12(180-frac180(n-2)n) \ angle BCA+angle ACD = angle BCD \ frac12(180-frac180(n-2)n)+120=frac180(n-2)n$$
This gives $n=9$.

share|cite|improve this answer

answered Jul 20 at 6:39

Piyush Divyanakar

3,258122

share|cite|improve this answer

share|cite|improve this answer

answered Jul 20 at 6:39

Piyush Divyanakar

3,258122

answered Jul 20 at 6:39

Piyush Divyanakar

3,258122

answered Jul 20 at 6:39

Piyush Divyanakar

3,258122

3,258122

add a comment | 

add a comment | 

 

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