How Can You Maximize the Area of a Triangle with Vertices on a Circle?

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Homework Help Overview

The discussion revolves around maximizing the area of a triangle with one vertex at the center of a circle of radius 1 and the other two vertices on the circumference. The problem involves geometric reasoning and understanding of triangle area formulas in relation to circular geometry.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore different methods to maximize the area, including placing vertices at specific coordinates and using angles. Questions arise regarding the correct formula for the area of a triangle and the implications of angle measures on area maximization.

Discussion Status

There is an ongoing exploration of the problem with various approaches being discussed. Some participants provide suggestions on how to set up the problem, while others question the validity of the formulas being used. No consensus has been reached, and the discussion remains active with multiple interpretations being considered.

Contextual Notes

Participants are navigating through potential misunderstandings regarding triangle area formulas and the relationship between angles and side lengths in the context of the problem. There is an emphasis on ensuring clarity in definitions and variables involved.

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What is the greatest possible area of a triangular region with one vertex at the center of radius 1 and the other two vertices on the circle?

What is the best way to maximize the area of a triangle in this instance?
 
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Put one vertex at (1,0), the second around at angle θ, calculate the corresponding area and maximize it as a function of θ.
 
The area of a triangle inscribed in circle is 1/2 * (AB)^{2} * sin(theta), correct? A 90 degree angle is going to yield maximum multiple of AB; however, is a lesser or greater angle with a longer length going to increase the area more?
 
What are A and B? The only variable in the problem is θ. And, no, that is not the correct formula for a triangle with sides A and B and included angle θ.
 
Then what is the formula for the area of a triangle using an angle theta?
 
The answer is 1/2, right?
 
You are reviewing for the GRE and you can't figure that out?? Draw a little triangle with sides A and B, included angle θ and height h and figure out its area.
 

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