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Math Help - easy equation problem

  1. #1
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    Cool easy equation problem

    not good at math so if you could please help it would be great.

    The sides of a triangle have these equations:
    y= -0.5x+1
    y= 2x-4
    y=-3x-9
    Verify that the triangle is an isosceles right triangle (without graphing)

    I've noticed that the second equation is -4 times the amount of the first if that helps
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  2. #2
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    Quote Originally Posted by Fox25 View Post
    not good at math so if you could please help it would be great.

    The sides of a triangle have these equations:
    y= -0.5x+1
    y= 2x-4
    y=-3x-9
    Verify that the triangle is an isosceles right triangle (without graphing)

    I've noticed that the second equation is -4 times the amount of the first if that helps
    find the points of intersection (x,y) to locate the vertices of the triangle, then find the distance between them using the distance formula.

    two distances should be equal, and all three distances should work in the Pythagorean Thm.
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  3. #3
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    thank you skeeter
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  4. #4
    Senior Member I-Think's Avatar
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    Greetings
    The sides of a triangle have these equations:
     y_1= -0.5x+1
     y_2= 2x-4
     y_3=-3x-9

    First things first, read over your notes, you wouldn't believe how much that helps.

    Now that you've done this (theoretically),let's tackle the question.
    To be an isosceles triangle, you need one line perpendicular to another and two lines to be equal to each other in length.

    In coordinate geometry, when one line is perpendicular to another, the gradient of the line is the negative reciprocal of the other.
    I.E. When the 2 gradients are multiplied, the product is negative 1.

    You may see now that y_1 gradient is the negative reciprocal of y_2, hence y_1 is perpendicular to y_2
    Hence the triangle formed by the intersection of the 3 lines is right-angled.

    Now, all we have to do is determine whether y_1 and y_2 are equal to each other. y_3 is opposite to the right angle, so it's the hypotenuse.

    Find where y_1 and y_2 intersect each other and y_3. Now with these coordinates the length of y_1 and y_2.

    y_1= -0.5x+1
      y_2= 2x-4
      y_3=-3x-9

    -0.5x+1=3x-9
    x=\frac{20}{7}
    Then y=\frac{-3}{7}
    A(\frac{20}{7},\frac{-3}{7})

    -0.5x+1=2x-4
    x=2
    Then y=0
    B(2,0)

    2x-4=3x-9
    x=5
    Then y=6
    C(5,6)

    You may graph at this point just to get your visual bearings.

    Now just apply the length of a line formula to determine that two of the lines are equal to each other.

    Edit: Be more like Skeeter and make it short
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