Okay, I've posted after a long time, I know... But I'm here with another doubt.

The ratio between the sides of 2 regular polygons is 4 : 5 and the ratio between the interior angles is 15 : 16. Find the number of sides.

Thanks!

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- Jan 6th 2007, 05:14 AMRuler of HellPolygons Help!
Okay, I've posted after a long time, I know... But I'm here with another doubt.

The ratio between the sides of 2 regular polygons is 4 : 5 and the ratio between the interior angles is 15 : 16. Find the number of sides.

Thanks! - Jan 6th 2007, 05:34 AMCaptainBlack
Let the number of sides of the polygons be $\displaystyle n$ and $\displaystyle m$, then the first condition tell you that:

$\displaystyle \frac{n}{m}=\frac{4}{5}\ \ \ \dots(1) $,

and as the interior angle of a ploygon with $\displaystyle n$ sides is $\displaystyle 180(n-2)/n$, the second condition tells us that:

$\displaystyle

\frac{n-2}{n}\,\frac{m}{m-2}=\frac{15}{16}

$

which may be simplified as we know $\displaystyle n/m=4/5$ to:

$\displaystyle

\frac{n-2}{m-2}=\frac{3}{4}\ \ \ \dots(2)

$

Now we can proceed to solve $\displaystyle (1)$ and $\displaystyle (2)$, or we can just guess, when we find $\displaystyle n=8$, $\displaystyle m=10$.

RonL - Jan 6th 2007, 05:50 AMRuler of Hell
Wait, I'm still a li'l confused. How did you solve equation 1 and 2? Sorry....

- Jan 6th 2007, 05:53 AMCaptainBlack
- Jan 6th 2007, 06:06 AMRuler of Hell
Got it! Yay, thanks a tonne Captain Black... (Though I used 2 different methods, yours and another one) Nonetheless, thanks!