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May 3rd, 2014, 02:34 PM   #1
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Geometry (triangle)

"In a triangle, all of the angles $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$ are less/lower than $\displaystyle 90°$. If all of the angles are integers, and the smallest angle is $\displaystyle \frac{1}{5}$ of the biggest angle, then what is the sum of the two biggest angles?"

I can't really figure this one out, as there seems to not be enough information.
I tried $\displaystyle a+b+c=180$. Lets say $\displaystyle a$ is the smallest angle and $\displaystyle c$ is the biggest. Then $\displaystyle a=\frac{c}{5}$

I then get $\displaystyle \frac{c}{5}+b+c=180$. If I solve this, I end up with
$\displaystyle 5b+6c=900$. I don't know how to go from here, but wolframalpha gives $\displaystyle b+c=\frac{b+900}{6}$.

Link to wolfram: (5b+6c=900 what is b+c - Wolfram|Alpha)

The answer is not supposed to be in this "algebra" way, it's supposed to be a whole number.

Does anyone know? The answer is supposed to be $\displaystyle b+c=163°$
If I know that $\displaystyle b+c=163°$, I can just add that into the equations, and I get $\displaystyle c=85°$, $\displaystyle b=78°$ and $\displaystyle a=17°$. But how am I supposed to find the solution without knowing the answer already ($\displaystyle b+c=163$)?

Thanks in advance =)
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May 3rd, 2014, 03:11 PM   #2
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We want to find $\displaystyle b+c$, where $\displaystyle a<b<c<90$ are the angles in the triangle.

We know that

$\displaystyle a+b+c=180$

so

$\displaystyle \frac{c}{5}+b+c=\frac{6c}{5}+b=180$

Since $\displaystyle b<c$, we know that

$\displaystyle \frac{6c}{5}+c=\frac{11c}{5}>180$

which gives

$\displaystyle c>81.818181...$

so now we have

$\displaystyle 81<c<90$

$\displaystyle c$ must be a multiple of $\displaystyle 5$, otherwise $\displaystyle a$ will not be an integer, hence $\displaystyle c=85$ and the other angles will be determined after.
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May 4th, 2014, 02:41 AM   #3
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Or (equivalently), as c = 5a, c < 90° implies 5a < 90°, and so a < 18°.
Also, as b = 180° - a - 5a, b < c implies 180° - 6a < 5a, and so a > 180°/11 = (16 4/11)°.
Hence a = 17°, which implies b =78° and c = 85°, and so b + c = 163°.
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