A regular dodecagon P_1, P_2, P_3, ... P_12 is inscribed in a circle with radius 1. Compute

(P_1 P_2)^2 + (P_1 P_3)^2 + ... + (P_11 P_12)^2. (The sum includes all terms of the form (P_i P_j)^2, where 1≤ i < j ≤ 12) I REMOVED LATEX

The (P_1 P_2) isn't multiplication, it shows the distance from point one to two.

I know this problem was posted before, but the solution is incorrect, so I would like to know how to solve it correctly.

MathsAreFun Mar 30, 2020

edited by
Guest
Mar 30, 2020

edited by Guest Mar 30, 2020

edited by Guest Mar 30, 2020

#1

#3**+1 **

I am still confused by the format.

What is (P_1 P_2)^2 supposed to mean? In other words, what is the operator inside the paranthesis? Scrutinize your question to make sure it is understandable.

Thank you.

AnExtremelyLongName
Mar 30, 2020

#4**0 **

Never mind! I am running on low-sleep. It is the length of the segments I suppose?

AnExtremelyLongName
Mar 30, 2020

#7**+1 **

Is this what it looks like?

\(\text{A regular dodecagon }P_1, P_2, P_3, ... P_12 \text{is inscribed in a circle with radius 1. }\)

\(\text{Compute } (P_1 P_2)^2 + (P_1 P_3)^2 + ... + (P_{11} P_{12})^2. \text{(The sum includes all terms of the form } (P_i P_j)^2, \text{where 1≤ i < j ≤ 12)}\)

In the series, are you sure the second term is \((P_1P_3)^2\)? It ruins the pattern!

AnExtremelyLongName Mar 30, 2020

#8

#9**0 **

I also don't think it is a series. It's supposed to be lengths from point to point within the dodecagon.

Guest Mar 30, 2020

#11**0 **

The problem cannot make sense as it ruins the pattern though.

So does it mean it includes EVERY diagonal and EVERY side length? Try to look at the original problem on the screen and match it with mine.

AnExtremelyLongName
Mar 30, 2020

#12**+2 **

**Hello Guest! ** I cannot correctly interpret the problem based on the format. I CAN give you a method to solve for one side length of the dodecagon, which no doubt can help you solve the problem yourself.

**Interpret the problem**

It's a pizza!

Look at ONE of the slices of the pizza. It has a vertex angle of 30 degress. (We calculate this by doing 180 - (360/12) = 150, then 180-150)

We know that the two long sides of the pizza is 1.

By law of cosines:

c^{2 }= a^{2} + b^{2} - 2ab*cos(C)

**Plug in and solve:**

c^{2} = 1 + 1 - 2cos(30)

c^{2 }= 2 - 2cos(30)

c^{2} = 1.691497100224

AnExtremelyLongName Mar 30, 2020

#13**+2 **

**A regular dodecahedron \(P_1 P_2 P_3 \dotsb P_{12}\) is inscribed in a circle with radius 1. 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 \((P_i P_j)^2\), where \(1 \le i < j \le 12\).**

**The \((P_1 P_2)\) isn't multiplication, it shows the distance from point one to two.**

See here: https://web2.0calc.com/questions/plshelp#r7

heureka Mar 30, 2020