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

[songoku]

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

 

[berkeman]

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.

 

[songoku]

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

 

[aheight]

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.

 

[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.

He has to show his correct solution before we can suggest any alternatives.

 

[songoku]

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

 

[aheight]

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.

 

[songoku]