Question 543318: I really need help with this please
distance between points (6, -2a) and (-6, 2a). Ans in simplified form
Thank you
Answer by lmeeks54(111)   (Show Source):
You can put this solution on YOUR website! In any of these, 'what is the distance between two points on a graph?' kind of problems, immediately think of the two points given as corner vertices of a right triangle, in which you know the two corner points connecting the hypotenuse.
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That's a mouthful; however, the point is, with two points of the form (x, y), you can draw the line between them as the hypotenuse (or long side of the triangle) and then imagine drawing in the base and height along the vertical (y) and horizontal (x) axes.
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You have two points: (6, -2a), (-6, 2a). Don't worry about the "a" term yet. Focus on getting the concept that these can always be imagined as part of a right triangle. And, recalling the Pythagorean Theorem, we know that the length of the hypotenuse of right triangle is solved by the general equation:
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c^2 = a^2 + b^2
where c = the hypotenuse and a and b = the base and height of the triangle.
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Bonus: if we calculate the length of the hypotenuse, we have solved your problem, to determine the length of the distance between the two points.
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Now we consider the "2a" part. No big deal, it is just the label for the length of the height of the triangle on the y dimension. In fact, to determine, the length of the base of the triangle (the part parallel to the x axis), we just need to subtract the two x coordinates:
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length of base = x1 - x2 ... 6 - -6 = 12. The base of the triangle = 12
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likewise, to determine the height of the triangle, we only need to subtract the two y coordinates:
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length of height = y1 - y2 ... -2a - 2a = -4a. In this case, we really only need the absolute value (the distance between them, not necessarily the positive or negative direction). You can keep the minus sign, but it will disappear when we square the term in our Pythagorean equation; or you can delete it now. In any case, the length of the height of the triangle is = 4a
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now we go into the equation for the right triangle and calculate for c, the hypotenuse, which is also the distance between our two points:
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c^2 = a^2 + b^2
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Let our height (h) = a
Let our base (b) = b
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c^2 = (4a)^2 + (12)^2
c^2 = 16a^2 + 144
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we can simplify this by taking the square root of both sides so we are solving for c instead of c^2; however, it doesn't make too much difference since we don't know "a", we can't go any further than that.
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c^2 = 16a^2 + 144
take square root of both sides:
c = sqrt(16a^2 + 144)
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This is the answer to your problem.
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cheers,
Lee
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