MHB Find the angle in the triangle

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The discussion revolves around finding the angle GDC in a triangle where BC is parallel to EF. Initial calculations suggest GDC equals 7x, and various angles are derived from the triangle's properties. The angles ADB, BDC, and ADC are discussed, leading to a realization that they sum to 360 degrees, correcting an earlier typo. The conclusion drawn is that x=10 degrees is the only solution that maintains valid triangle angles, with the possibility of multiple integer solutions for x between 0 and 180 degrees. The challenge remains to identify a specific angle that would confirm the solution's validity.
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I've tried:
BC || EF
How to find angle GDC? I think GDC=7x but why?
I have an answer but how to solve this?
 
Last edited:
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maxkor said:
https://www.physicsforums.com/attachments/11871
View attachment 11873

I've tried:
BC || EF
How to find angle GDC? I think GDC=7x but why?
I have an answer but how to solve this?
There's another approach. Look at angle BAC. It's 180 - 7x - 7x. (ABC is isosceles, but we don't need this.) So we can get angle DAC in terms of x.

Now find the angles ADB, BDC, and ADC. They sum to 180...

-Dan
 
BAC=180-15x
ADB=180-3x
BDC=180-8x
ADC=11x
What next?
 
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Check that again. BAC = 180 - 14x.

ADB + BDC + ADC= 180. Why?

-Dan
 
ok, BAC = 180 - 14x
Can you tel why ADB + BDC + ADC= 180? I don't see it.
 
maxkor said:
ok, BAC = 180 - 14x
Can you tel why ADB + BDC + ADC= 180? I don't see it.
Sorry, that was a typo. They sum to 360. Notice that all three angles sum to form a circle.

-Dan
 
ADB=180-3x
BDC=180-8x
ADC=11x
ADB + BDC + ADC=180-3x+180-8x+11x=360
And what next??
 
maxkor said:
ADB=180-3x
BDC=180-8x
ADC=11x
ADB + BDC + ADC=180-3x+180-8x+11x=360
And what next??
Well, it made a lot more sense when I had the typo! :)

I'll have to get back to you on this. Sorry about that!

-Dan
 
Picture to scale with $x=0.5^\circ$ where the angles are correct except for $\angle ACD$.
\begin{tikzpicture}
\def\x{0.5}
\def\d{4}
\def\b{\d * sin(3*\x) / sin(2*\x)}
\def\a{(\b) * sin(14*\x) / sin(7*\x)}
\def\e{(\b) * sin(\x) / sin(3 * \x)}
\coordinate[label=left:B] (B) at (0,0);
\coordinate[label=A] (A) at ({7*(\x)}:{\b});
\coordinate[label=D] (D) at ({5*\x}:{\e});
\coordinate[label=right:C] (C) at ({\a},0);
\draw (A) -- (B) -- (C) -- cycle (B) -- (D) -- (C) (D) -- (A);
\end{tikzpicture}
Note that $\angle ACD=4x$ is too big in the picture.
Btw, this is TikZ code in which we can change the angle $x$ if we want to play with it.

Solution

The angles at B and C are both $7x$. Therefore the triangle is isosceles, and
$$AB=AC$$

From the law of sines:
$$\triangle ABD:\quad\frac{AB}{\sin(180^\circ-3x)}=\frac{AD}{\sin(2x)}$$
$$\triangle ACD:\quad\frac{AC}{\sin(11x)}=\frac{AD}{\sin(4x)}$$

Combine to find:
$$\frac{AB}{AD}=\frac{\sin(3x)}{\sin(2x)}=\frac{\sin(11x)}{\sin(4x)} \implies \sin(3x)\sin(4x)=\sin(2x)\sin(11x)\tag 4$$

Wolfram seems to say that $x=0$.

EDIT: with some twiddling, $x=10^\circ$ rolls out as the only solution in which we actually have a triangle.
 
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  • #11
maxkor said:
Ok but answer is 10 not 0, so how solve this without wolfram.
I've tried to find a trick solution.
Interesting, Wolfram confirms that $x=10^\circ$ is also a solution of the same equation. It appears that I misinterpreted the output of Wolfram.

\begin{tikzpicture}
\def\x{10}
\def\d{4}
\def\b{\d * sin(3*\x) / sin(2*\x)}
\def\a{(\b) * sin(14*\x) / sin(7*\x)}
\def\e{(\b) * sin(\x) / sin(3 * \x)}
\coordinate[label=left:B] (B) at (0,0);
\coordinate[label=A] (A) at ({7*(\x)}:{\b});
\coordinate[label=D] (D) at ({5*\x}:{\e});
\coordinate[label=right:C] (C) at ({\a},0);
\draw (A) -- (B) -- (C) -- cycle (B) -- (D) -- (C) (D) -- (A);
\end{tikzpicture}
 
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  • #12
But that equation is hard to solve.
 
  • #13
This is the problem with the 4 point kangaroo competition.
 
  • #14
maxkor said:
This is the problem with the 4 point kangaroo competition.
There doesn't seem to be a problem 1469 in there:

1658234890038.png
 
  • #15
maxkor said:
I've tried to find a trick solution.

If only we could prove that $\triangle CGD$ is similar to $\triangle AGC$. 🤔
We have 1 angle that is the same.
We either need another angle, or we need to prove $\triangle CGD$ is also isosceles.
 
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  • #16
Look page 74.
 
  • #17
Klaas van Aarsen said:
If only we could prove that $\triangle CGD$ is similar to $\triangle AGC$. 🤔
We have 1 angle that is the same.
We either need another angle, or we need to prove $\triangle CGD$ is also isosceles.
$\triangle CGD$ is not also isosceles.
If only we could prove that $\triangle CGD$ is similar to $\triangle AGC$, if GDC=7x ??
 
  • #18
Turns out that there are $13$ integer solutions for $x$ in degrees between $0^\circ$ and $180^\circ$. And those are the only solutions.
The solution $x=10^\circ$ is the only one such that all angles of the triangle are less than $180^\circ$.

The interesting part is that angles with an integer number of degrees are solutions.
For that to make sense, there must be angles hidden somewhere of a multiple of $x$ that is $90^\circ$ so that the cosine is zero, or $180^\circ$ so that the sine is zero.

Put otherwise, if we can find an angle of $9x$ somewhere (or $18x$), and if we can prove that it is a right angle (respectively straight angle), then the solution $x=10^\circ$ comes rolling out.
Unfortunately I haven't found an angle of $9x$ yet, although I do have $x,2x,3x,4x,5x,7x,8x,10x,11x,12x$ and their complements.
 
  • #19
Maybe you have GDC=7x? we at home :)
 
  • #20
sol.png
 
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