Finding maximal area of triangle.

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Homework Help Overview

The problem involves finding the point R on the unit circle that maximizes the area of triangle PQR, where P(1,0) and Q(0,1) are fixed points. The context is rooted in geometry and calculus, particularly in relation to triangle area calculations using vectors.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss using vectors to calculate the area of the triangle and mention the determinant form for area calculation. There is uncertainty about how to approach the problem, with one participant suggesting a guess involving specific coordinates for R. Others express confusion regarding the use of the cross product, indicating a lack of familiarity with that concept.

Discussion Status

The discussion is ongoing, with participants exploring different methods to approach the problem. Some guidance has been provided regarding vector calculations and area formulas, but there is no consensus on the best method or solution yet.

Contextual Notes

One participant notes that the cross product has not been covered in their course, which may limit their ability to apply certain suggested methods. There is also mention of the problem being part of an integration question set, which may impose additional constraints on the approach.

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Homework Statement



Consider the points P(1,0) and Q(0,1) on the unit circle. For which point R(cos(theta),sin(theta)) on the unit circle, is the area of the triangle PQR maximal?

Homework Equations



Hint - It might be easier to use vectors to calculate the area of the triangle.

The Attempt at a Solution



I'm really not sure how to approach this question. It's part of an integration question set. A wild guess would assume cos(pi/4) and sin(pi/4) would be the co-ordinates that would give a maximal area but I'm not sure if that's right or how to prove it.

Thanks for any help.
 
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KaliBanda said:

Homework Statement



Consider the points P(1,0) and Q(0,1) on the unit circle. For which point R(cos(theta),sin(theta)) on the unit circle, is the area of the triangle PQR maximal?

Homework Equations



Hint - It might be easier to use vectors to calculate the area of the triangle.

The Attempt at a Solution



I'm really not sure how to approach this question. It's part of an integration question set. A wild guess would assume cos(pi/4) and sin(pi/4) would be the co-ordinates that would give a maximal area but I'm not sure if that's right or how to prove it.

Thanks for any help.

Guesses don't count, even if they are wild. Make vectors out of two sides of the triangle and remember that the area is ##\frac 1 2|\vec A \times \vec B|##. Use a little calculus on that. You can make the third component of the vectors 0 to use the cross product.

[Edit] I should have mentioned the determinant form for the area of a triangle given its vertices in case you haven't had 3D vectors. If the vertices are ##(a,b),(c,d),(e,f)## the area is also the absolute value of the determinant$$
A = \frac 1 2\left |\begin{array}{ccc}
1 & a & b\\

1 & c & d\\

1 & e & f
\end{array}\right|$$
 
Last edited:
LCKurtz said:
Guesses don't count, even if they are wild. Make vectors out of two sides of the triangle and remember that the area is ##\frac 1 2|\vec A \times \vec B|##. Use a little calculus on that. You can make the third component of the vectors 0 to use the cross product.

We haven't learned the cross product as it isn't part of our couse, but I'll give it a go.

Thanks for the reply!
 
KaliBanda said:
We haven't learned the cross product as it isn't part of our couse, but I'll give it a go.

Thanks for the reply!

I edited my post with another 2D suggestion.
 

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