Circles and Angles

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homeschool girl

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The question:

A regular dodecagon [MATH]P_1 P_2 P_3 \dotsb P_{12}[/MATH] is inscribed in a circle with radius [MATH]1.[/MATH] Compute
\[(P_1 P_2)^2 + (P_1 P_3)^2 + \dots + (P_{11} P_{12})^2.\]
(The sum includes all terms of the form [MATH](P_i P_j)^2,[/MATH] where [MATH]1 \le i < j \le 12.[/MATH])

Where I'm stuck:
I drew out the dodecagon and labeled all the points, I figured out how to find lines like [MATH]\overline{P_1P_4}[/MATH] that form right angles and are equal to [MATH] \sqrt{r^2+r^2}= \sqrt{1+1}=\sqrt{2}[/MATH], and lines like [MATH]\overline{P_1P_7}[/MATH] that form straight lines and are equal to [MATH] r+r = 2[/MATH], but I'm unsure on how to find the other types of lines.
 
homeschool girl said:
The question:

A regular dodecagon [MATH]P_1 P_2 P_3 \dotsb P_{12}[/MATH] is inscribed in a circle with radius [MATH]1.[/MATH] Compute
\[(P_1 P_2)^2 + (P_1 P_3)^2 + \dots + (P_{11} P_{12})^2.\]
(The sum includes all terms of the form [MATH](P_i P_j)^2,[/MATH] where [MATH]1 \le i < j \le 12.[/MATH])

Where I'm stuck:
I drew out the dodecagon and labeled all the points, I figured out how to find lines like [MATH]\overline{P_1P_4}[/MATH] that form right angles and are equal to [MATH] \sqrt{r^2+r^2}= \sqrt{1+1}=\sqrt{2}[/MATH], and lines like [MATH]\overline{P_1P_7}[/MATH] that form straight lines and are equal to [MATH] r+r = 2[/MATH], but I'm unsure on how to find the other types of lines.
Click to expand...
Draw the circle that circumscribes the polygon. Let the center be O.

What is the m<(1O2) = ?

Do you know the Law of cosines for triangles? use it to calculate Length of PnPn+1
 
homeschool girl said:
also, what does m stand for?
Click to expand...
The expression [MATH]m\angle(P_1OP_2)[/MATH] means "the measure of angle [MATH]P_1OP_2[/MATH]". But that isn't important if you won't be doing trigonometry.

homeschool girl said:
Can I please have help with the problem?
Click to expand...
I had to draw my own picture and ponder it a bit.

You're sort of on the right track when you say,
homeschool girl said:
I figured out how to find lines like [MATH]\overline{P_1P_4}[/MATH] that form right angles and are equal to [MATH] \sqrt{r^2+r^2}= \sqrt{1+1}=\sqrt{2}[/MATH], and lines like [MATH]\overline{P_1P_7}[/MATH] that form straight lines and are equal to [MATH] r+r = 2[/MATH], but I'm unsure on how to find the other types of lines.
Click to expand...
though those are special cases.

Did you notice that every segment [MATH]P_iP_j[/MATH] pairs up with another perpendicular to it? Here I matched up every segment from P_1 with another (same color) that does this. What happens when you add their squares?

1607739688352.png

By the way, it would be helpful if you would tell us what topic a problem was under, to give us a hint about what sort of math you're expected to use. What was it in this case?
 
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