Find Slope, Equation, MidpointDate: 06/04/2003 at 23:50:08 From: Ashley Subject: Slopes A = (-4,6) and B = (-2,1) 1) find the slope; 2) find the equation of the line containing A and B; 3) find the coordinates for the midpoint AB; 4) find the distance between A and B. Date: 06/05/2003 at 09:57:01 From: Doctor Dotty Subject: Re: Slopes Hi Ashley, Thanks for the question. Any (non-vertical) line can be represented by the equation: y - y1 = m(x - x1) Where m is the gradient (slope) and two points it passes through are (x, y) and (x1, y1). Let's think about a line passing through (2, 4) and (-3, 6) as an example. We have two points, so x = 2, y = 4; x1 = -3, y1 = 6. This gives: 4 - 6 = m(2 - -3) -2 = 5m m = -2/5 So the gradient (slope) is -2/5. Next we need an equation of the line. We can find this using one point and the gradient. So: y - y1 = m(x - x1) y - 6 = (-2/5)(x - -3) y - 6 = (-2/5)(x + 3) Multiply by 5: 5y - 30 = -2(x + 3) 5y - 30 = -2x - 6 5y + 2x - 24 = 0 Which is the equation of the line. Now, the midpoint. That is the point that lies horizontally halfway between x and x1, and vertically halfway between y and y1. To remind us: x = 2, y = 4; x1 = -3, y1 = 6. Halfway between 2 and -3 is -0.5. Halfway between 4 and 6 is 5. Therefore the coordinates of the midpoint are (-0.5, 5). For easier use in the future, you can write this method as: x + x1 y + y1 Midpoint is at ------ , ------ 2 2 Now, the distance from A to B. . | . | A 6 + | . | | | | | . |_ _ _ 4_+_ _ _.B | . | . | . | . -----------+--------+------+--------------- -3 | 2 . | | Let's look at that triangle: A | . d 2| |_ . |_|_ _ _ _ _ _ .B 5 Using Pythagoras' theorem, 2^2 + 5^2 = d^2 (where ^2 means squared) 4 + 25 = d^2 sqrt(29) = d Which is the distance between the two points. For easier use in the future, you can write this method as an equation: d = sqrt[ (x - x1)^2 + (y - y1)^2 ] Can you apply this to your question now? Write back if I can be of any more help - on this or anything else. - Doctor Dotty, The Math Forum http://mathforum.org/dr.math/ |
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