Angular Coeficient of a Line Passing Through (4,3) in 1st Quadrant with Area 27

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The discussion focuses on finding the angular coefficient of a line passing through the point (4,3) that forms a triangle with the coordinate axes, having an area of 27 square units. The area of the triangle is expressed as half the product of its base and height, which can be derived from the line's x and y intercepts in terms of the slope (a). By setting the area equation equal to 27 and solving for the slope, two possible values for the angular coefficient are determined: a = -1.5 and a = -0.375. The solution process emphasizes the relationship between the line's intercepts and the area of the triangle formed with the axes. This approach effectively resolves the problem using geometric principles.
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Homework Statement


What is the angular coeficient of a straight line that passes through (4,3), if it satifies x>0 and y>0, and it is one of the sides of a triangle with total area of 27 (units of area).

calcule o coeficiente angular de um reta que passe por (4,3) de modo que a parte da reta no primeiro quadrante forma um triângulo de área 27 com os eixos coordenados positivos.


Homework Equations


I know that the area of a triangle is area = base * height/2 . I know the equation of the straigh line is y = aX + B, where "a" is what I am looking for. But I am stucked! I can't find a nice relation.



The Attempt at a Solution



I've tried to use the relation Area = 1/2 * D
Where D is the determinant of a matrix with the 3 vertices of the triangle, with an aditional columns of "1". But, couldn't go any further... any ideas?
 
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amarante said:

Homework Statement


What is the angular coeficient of a straight line that passes through (4,3), if it satifies x>0 and y>0, and it is one of the sides of a triangle with total area of 27 (units of area).

calcule o coeficiente angular de um reta que passe por (4,3) de modo que a parte da reta no primeiro quadrante forma um triângulo de área 27 com os eixos coordenados positivos.


Homework Equations


I know that the area of a triangle is area = base * height/2 . I know the equation of the straigh line is y = aX + B, where "a" is what I am looking for. But I am stucked! I can't find a nice relation.



The Attempt at a Solution



I've tried to use the relation Area = 1/2 * D
Where D is the determinant of a matrix with the 3 vertices of the triangle, with an aditional columns of "1". But, couldn't go any further... any ideas?

I assume the axes form the other two sides of the triangle. Write the equation of the line though ##(4,3)## with slope ##a##. Its ##x## and ##y## intercepts will give you the base and height in terms of ##a##. Then set the area, now in terms of ##a##, equal to 27 and solve for ##a##.
 
Perfect! I got it. Thank you so much!

I got a solution with two possibilities, a= -1.5, or a = -0.375.

Thanks!
 

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