 Posers and Puzzles

1. 16 Mar '07 04:471 edit
http://i160.photobucket.com/albums/t166/ginoj_2007/Geometry.jpg

This is the question. We need to find the A angle.

(For ppl who cannot read: The B and C angles are equal; [BC]=[AD]=1 ; [DC]=Square Root 2.)
2. 17 Mar '07 06:291 edit
Originally posted by GinoJ
http://i160.photobucket.com/albums/t166/ginoj_2007/Geometry.jpg

This is the question. We need to find the A angle.

(For ppl who cannot read: The B and C angles are equal; [BC]=[AD]=1 ; [DC]=Square Root 2.)
from C drop a perpendicular to AB such that you have a right angled triangle with side lengths 2, 1, and AC of length sqrt(5), then just calculate asin(1/sqrt(5))
3. 18 Mar '07 22:48
36 deg
4. 19 Mar '07 04:17
Originally posted by Agerg
from C drop a perpendicular to AB such that you have a right angled triangle with side lengths 2, 1, and AC of length sqrt(5), then just calculate asin(1/sqrt(5))
How do you know that the length of AC is sqrt(5)?
5. 19 Mar '07 11:51
Originally posted by DeepThought
How do you know that the length of AC is sqrt(5)?
A^2+B^2=C^2

6. 19 Mar '07 13:371 edit
Originally posted by strokem1
A^2+B^2=C^2

Ok. then why is the other length 2?

I'm sorry, but unless there is some rule that tells you what the length of the perpendicular is in a right angled triangle constructed in the manner described then the solution to the problem isn't as straightforward as is being suggested.

Nowhere in the diagram does it suggest that the angles or are right angles, and in any case the length DC is sqrt(2) and not 2. How do you drop a perpendicular from C to AB and prove that the lengths are those given - they may well be but I'd like to know why.

I think that the key to this problem is to do with the fact that ADC + ADB = Pi so that you can use the sine formulae to relate all the lengths to one another, and to the angles and solve for A. Which I will attempt once I've had some lunch.
7. 19 Mar '07 16:452 edits
Here is a partial solution:

The length BC = 1, the length AB = AC = x and is unknown.

Applying the cosine formula (http://en.wikipedia.org/wiki/Law_of_cosines) to the large triangle gives:

1 = 2*x^2 - 2 x^2 Cos(A), (1)

Applying it to the triangle ADC gives:

2 = 1 + x^2 - 2*x Cos(A), (2)

Eliminating Cos(A) we get.

x^3 - 2x^2 - x + 1 = 0 (3)

Solve equation 3 using a symbolic manipulation program (too much of a pain to do by hand) and substitute the largest root into 1 or 2 to get the answer.

We know that x = AB is greater than AD = 1. The cubic equation y = x^3 - 2x^2 - x + 1 has turning points at x = (2 +/- sqrt(7))/3 so has three real roots. y = 1 for x = 0, for x large the cubic term dominates so y is negative for large negative x and positive for large positive x. For x = 1, y = 1 - 2 - 1 +1 = -1 so the roots of equation 3 are negative, between 0 and 1 and greater than 1. We want the root that is larger than 1.

Edit: Solving (3) numerically gives AB = 2.24698 and the angle A as 0.448799 radians or 25.714 degrees.

Edit 2: Note that 2.24698 != sqrt(5) = 2.236
8. 21 Mar '07 05:44
Originally posted by DeepThought
Here is a partial solution:

The length BC = 1, the length AB = AC = x and is unknown.

Applying the cosine formula (http://en.wikipedia.org/wiki/Law_of_cosines) to the large triangle gives:

1 = 2*x^2 - 2 x^2 Cos(A), (1)

Applying it to the triangle ADC gives:

2 = 1 + x^2 - 2*x Cos(A), (2)

Eliminating Cos(A) we get.

x^3 - 2x^2 - x + 1 = ...[text shortened]... e angle A as 0.448799 radians or 25.714 degrees.

Edit 2: Note that 2.24698 != sqrt(5) = 2.236
Imppressive!

Yes, the answer is 180/7 degrees.
9. 21 Mar '07 13:40
Originally posted by DeepThought
Here is a partial solution:

The length BC = 1, the length AB = AC = x and is unknown.

Applying the cosine formula (http://en.wikipedia.org/wiki/Law_of_cosines) to the large triangle gives:

1 = 2*x^2 - 2 x^2 Cos(A), (1)

Applying it to the triangle ADC gives:

2 = 1 + x^2 - 2*x Cos(A), (2)

Eliminating Cos(A) we get.

x^3 - 2x^2 - x + 1 = ...[text shortened]... e angle A as 0.448799 radians or 25.714 degrees.

Edit 2: Note that 2.24698 != sqrt(5) = 2.236
Could you walk me through how you got from equation (2) to equation(3). I can't follow how you eliminated cos(A) or how you end up with a cubic equation.
10. 21 Mar '07 17:24
Originally posted by luskin
Could you walk me through how you got from equation (2) to equation(3). I can't follow how you eliminated cos(A) or how you end up with a cubic equation.
Rearrange equation (1) to give:

2 x^2 Cos(A) = 2*x^2 - 1, (A)

do the same with equation (2) to give:

2 x Cos(A) = x^2 - 1,

therefore (this is probably the step you missed - leaving out the power of two wasn't a typo)

2 x^2 Cos(A) = x^3 - x (B)

the left hand sides of A and B are equal and so:

x^2 - 1 = x^3 - x

Which rearranges to give equation 3 from my post above.

I thought I'd got it wrong at first because the numbers didn't come out to anything neat, which they normally do with this type of problem. I didn't like the look of what comes from substituting into the general equation for the roots of a cubic equation so went for Newton Rapheson instead. I'd be interested to know if there's a way of getting to the answer analytically.
11. 21 Mar '07 22:25
Originally posted by DeepThought
Rearrange equation (1) to give:

2 x^2 Cos(A) = 2*x^2 - 1, (A)

do the same with equation (2) to give:

2 x Cos(A) = x^2 - 1,

therefore (this is probably the step you missed - leaving out the power of two wasn't a typo)

2 x^2 Cos(A) = x^3 - x (B)

the left hand sides of A and B are equal and so:

x^2 - 1 = x^3 - x

Which rearranges to ...[text shortened]... nstead. I'd be interested to know if there's a way of getting to the answer analytically.
Thanks for that. I see it all now. Yes it would be interesting to see any other way of getting to it. I thought from the look of the diagram that it probably involved the golden ratio somehow. But that line kept leading to dead ends and wrong answers.
12. 22 Mar '07 00:384 edits
Originally posted by DeepThought
How do you know that the length of AC is sqrt(5)?
was up to no good here, sorry.
13. 31 Mar '07 12:11
Originally posted by DeepThought
Here is a partial solution:

The length BC = 1, the length AB = AC = x and is unknown.

Applying the cosine formula (http://en.wikipedia.org/wiki/Law_of_cosines) to the large triangle gives:

1 = 2*x^2 - 2 x^2 Cos(A), (1)

Applying it to the triangle ADC gives:

2 = 1 + x^2 - 2*x Cos(A), (2)

Eliminating Cos(A) we get.

x^3 - 2x^2 - x + 1 = ...[text shortened]... e angle A as 0.448799 radians or 25.714 degrees.

Edit 2: Note that 2.24698 != sqrt(5) = 2.236
Apologies folks...about half an hour after submitting that post of mine I realised I'd made the tragic mistake of jumping to a wrong conclusion from my diagram and accepting something without justifying it!...I couldn't get back to edit the post neither 🙁
14. 02 Apr '07 09:18
I had always thought compass and straight edge weren't enough to make a regular heptagon..

However, in trying to use this figure to craft one, the 3 segments (and identical angle) are too disjointed to be contructed without further aid.

Yikes on the math..
15. 23 Sep '07 13:44
Originally posted by Agerg
from C drop a perpendicular to AB such that you have a right angled triangle with side lengths 2, 1, and AC of length sqrt(5), then just calculate asin(1/sqrt(5))
That can't be correct. The length of the perpendicular from C on AB has to be less than 1. Let the foot of the perpendicular from C on AB be P;

Then obviously CP < BC and BC =1. Hence your assumption about DP being equal to 1 and the right angled triangle APC having sides of length 2, 1 and AC is wrong. Hence AC cannot be sqrt(5).