Finding the Sum of Perpendiculars in a 24-Sided Polygon Inscribed in a Circle

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Homework Help Overview

The problem involves a regular 24-sided polygon inscribed in a circle, where perpendiculars are dropped from the vertices to a diameter. The goal is to find the sum of the lengths of these perpendiculars.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the absence of a given formula and explore potential trigonometric identities relevant to the problem. There is an attempt to express the lengths of the perpendiculars in terms of the radius of the circle and the angles associated with the polygon's vertices.

Discussion Status

Some participants have suggested using trigonometric functions to express the lengths of the perpendiculars, while others are questioning the initial assumptions and the formulas that could be applied. There is ongoing exploration of how to sum the sine values of the angles corresponding to the vertices.

Contextual Notes

Participants note that the problem is situated within a section on trigonometric identities and series, indicating a potential reliance on those concepts for solving the problem.

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Homework Statement



There is a regular 24 sided polygon inscribed in a circle. A diameter is drawn and perpendiculars are dropped from all the vertices on that side of the diameter. Find the sum of the lengths of the perpendiculars.

The Attempt at a Solution



I came across this problem and I have no clue how to approach it. I'm looking for a point in the right direction. Thanks for the help!
 

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hi armolinasf!

well, what formula are you starting with? :smile:
 
No formula was given. It was at the end of a section on trig identities, specifically using them to sum trig series.
 
You have to find the formula. You have studied trigonometry haven't you?

ehild
 
My first instinct was to write something like S=r+2(l1+l2+l3+l4+l5), where l is the length of each chord. Should I be looking for a way to express each l in terms of r?
 
hi armolinasf! :smile:

yes, and the various angles (nπ/12)
 
Yeah i finally realized that the height of each perpendicular is equal to the sine of the angles...So here my sum:

S=r+2r[sin(pi/12)+sin(pi/6)+sin(pi/4)+sin(pi/3)+sin(5pi/12)]
 
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