How to Find the Ratio of Areas in a Scalene Triangle?

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SUMMARY

The discussion focuses on finding the ratio of areas in a scalene triangle using a specific formula from Sciencing. The user describes their approach involving the area of triangle ABC and the area of triangle OBC, utilizing variables such as p, q, r, s, u, and v. The method employed is characterized as brute force, indicating a complex solution process. Participants emphasize the need for the user to share their solution for further simplification suggestions.

PREREQUISITES
  • Understanding of triangle area formulas, specifically for scalene triangles.
  • Familiarity with algebraic manipulation of equations.
  • Basic knowledge of coordinate geometry and vertex assignments.
  • Proficiency in using LaTeX for mathematical formatting.
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  • Research the formula for calculating the area of a triangle given its vertices.
  • Learn about coordinate geometry and its applications in solving triangle area problems.
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Students studying geometry, mathematics educators, and anyone interested in solving complex area problems in triangles.

songoku
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Homework Statement
The vertices of triangle ABC are A (p,q), B (r, s), C (u, v). Given that those points fulfill equation 2p + 3r + 4u = 0 and 2q + 3s + 4v = 0 and the origin lies in the interior of the triangle. If area of OBC = r times of ABC, find r
Relevant Equations
Not sure
I do not know where to start. I draw scalene triangle and assign each coordinates to the vertices. Tried writing something but none working. Please give me a hint to start.

Thanks
 
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Can you attach a PDF or JPEG file with your sketch in it please? And please show as much of your work as you can. Thanks.
 
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berkeman said:
Can you attach a PDF or JPEG file with your sketch in it please? And please show as much of your work as you can. Thanks.

I think I get it. Thank you very much berkeman
 
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songoku said:
I think I get it. Thank you very much berkeman
Am I to understand you solved it and are done? If so then I would like to know if there is a simpler way of solving it than what I did. My solution is not simple. I can show my work if that is not interfering with forum policies.
 
aheight said:
Am I to understand you solved it and are done? If so then I would like to know if there is a simpler way of solving it than what I did. My solution is not simple. I can show my work if that is not interfering with forum policies.
He has to show his correct solution before we can suggest any alternatives. :smile:
 
aheight said:
Am I to understand you solved it and are done? If so then I would like to know if there is a simpler way of solving it than what I did. My solution is not simple. I can show my work if that is not interfering with forum policies.

Yes I solved it but I don't think my method is simple. I use formula from this link https://sciencing.com/area-triangle-its-vertices-8489292.html

I set up equation using information area ABC = r . Area OBC (the equation has a lot variables p, q, r, s u, v) then using another two equations given from question (I make p and q as subject) I substitute them to equation of area. Basically I use brute force to solve it 😅
 
songoku said:
I set up equation using information area ABC = r . Area OBC (the equation has a lot variables p, q, r, s u, v) then using another two equations given from question (I make p and q as subject) I substitute them to equation of area. Basically I use brute force to solve it 😅

Kindly provide a summary of your work including a sample plot with labels. Nicely formatted in Latex would be nice too.:cool:
 
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Document_1.jpg
 

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