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Math Help - Area of a triangle. Formed by two axes and a line. Small question.

  1. #1
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    Area of a triangle. Formed by two axes and a line. Small question.

    Hello everyone,

    Heres the problem; A right triangle is formed in the first quadrant by the x and y axis and a line through point (2,1). Write the area A of the triangle as a function of x.

    Using the information above I can conjure these three points;
    (0,y)
    (2,1)
    (x,0)

    Furthermore, I know area of a triangle is equal to .5(bxh) or in this case (y*x)/2.

    The problem is that I need to express this area as a function of x. Which sounds simple. Unfortunately, that means I need to get y in terms of x. This is where I am stuck.

    How would I put y in terms of x?

    Thanks,
    Taylor S. Amarel
    Living is learning.
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  2. #2
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    Quote Originally Posted by tsa256 View Post
    Hello everyone,

    Heres the problem; A right triangle is formed in the first quadrant by the x and y axis and a line through point (2,1). Write the area A of the triangle as a function of x.

    Using the information above I can conjure these three points;
    (0,y)
    (2,1)
    (x,0)

    Furthermore, I know area of a triangle is equal to .5(bxh) or in this case (y*x)/2.

    The problem is that I need to express this area as a function of x. Which sounds simple. Unfortunately, that means I need to get y in terms of x. This is where I am stuck.

    How would I put y in terms of x?

    Thanks,
    Taylor S. Amarel
    Living is learning.
    You can split the triangle up into two "internal" right-triangles and a rectangle.

    The rectangle area is 2(1)=2 square units.

    The area of the inner triangle to the right of the rectangle is

    \displaystyle\frac{(x-2)(1)}{2}

    The area of the inner triangle above the rectangle is

    \displaystyle\frac{(y-1)(2)}{2}

    Then, writing y as a function of x....

    \displaystyle\frac{y-1}{0-2}=\frac{1-0}{2-x}

    from the slope of the line.

    If you use that relationship to write y in terms of x,
    then you can write the triangle area in terms of x only, without y.

    Of course, then you can ignore the partitioning of the triangle!
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  3. #3
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    area = xy / 2

    equating the slopes: y = x / (x-2)

    area = x[x / (x-2)] / 2 = x^2 / [2(x-2)]
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