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The Geometry Forum Newsletter

Summer 1994, page 2

Join us online: POW -- Problem of the Week

The Problem of the Week is a regular feature here at the Geometry Forum. Each weekend a high school level geometry problem is posted, and the following weekend a summary of solutions and their authors is posted.

Problem of the Week for April 4-8

Joaquin and Juanita go to the movies. Joaquin says it's best to sit so that the angle formed by the line of sight between the left and right sides of the screen is 30 degrees, and that there's only one seat in the theater for which this is true. Juanita says he's wrong -- there are other places where the viewing angle is 30 degrees.

Who's right? Why?

The solutions this week are interesting. Some folks have obviously studied the "implied" theorem. Others reasoned out correct answers. Correct solutions were submitted by Hilary Aleksa and Allison Sullivan.

From: PDALEY@fair1.fairfield.edu

Solution from Hilary Aleksa, grade 9, Fairfield High School
Using the movie screen as the base, draw a triangle to where Joaquin thinks the best seat is. Then draw a circle so that the triangle is inscribed in the circle. The screen then becomes a chord of the circle. Therefore, the 30-degree angle intercepts the arc which corresponds with the screen (chord). Any other angle formed in the theater that intercepts that same arc is also 30 degrees. Juanita, as a result, is the correct one; there is more than one seat meeting those qualifications.

Solution from Allison Sullivan, grade 9, Fairfield High School
Juanita is right, because if a circle is drawn around the picture given, so that the screen is a chord of the circle, then all the inscribed angles intercepting this chord (the screen) will equal 30 degrees. Therefore, there is more than 1 seat that will have a line of sight of 30Ľ.

Theorem: If inscribed angles intercept the same arc, then the inscribed angles are congruent.
If angles intercept the same arc, then they intercept the same chord.

Therefore, all the inscribed angles that have the screen as an intercepted chord have the measure of 30 degrees.

Every seat lying in front of the screen on the circle that circumscribes triangle DEG has a viewing angle of 30 degrees. The best seat can be found not by looking for a 30-degree line of sight (seats near D or E will give a distorted view of the screen) but by finding a point at the intersection of the circle and a line passing perpendicularly through the midpoint of DE. Triangle DEG will then be an isosceles triangle with G equidistant from D and E.

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