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Lines and Distance Lines and distance are fundamental to coordinate geometry, not to mention to the Math IIC test. Even the most complicated coordinate geometry question will use the concepts covered in the next couple ofsections. Distance Measuring distance in the coordinate plane is made possible thanks to the Pythagorean theorem. If you are given two points, (x1y1) and (x2y2), their distance from each other is given by the following formula: (http://img.sparknotes.com/content/testprep/bookimgs/sat2/math2c/0006/distance=(x2-x1)2+(y2-y1)2+.gif) The diagram below shows how the Pythagorean theorem plays a role in the formula. The distance between two points can be represented by the hypotenuse of a right triangle whose legs are of lengths (x2 – x1) and (y2 – y1). (http://img.sparknotes.com/content/testprep/bookimgs/sat2/math2c/0006/distance.gif) To calculate the distance between (4, –3) and (–3, 8), plug the coordinates into the formula: (http://img.sparknotes.com/content/testprep/bookimgs/sat2/math2c/0006/distance=((-3)4)2.gif) The distance between the points is (http://img.sparknotes.com/content/testprep/bookimgs/sat2/math2c/0007/m2c.total287.gif), which equals approximately 13.04. You can double-check this answer by plugging it back into the Pythagorean theorem. Finding Midpoints The midpoint between two points in the coordinate plane can be calculated using a formula. If the endpoints of a line segment are (x1y1) and (x2y2), then the midpoint of the line segment is: (http://img.sparknotes.com/content/testprep/bookimgs/sat2/math2c/0053/midpoint-(x1+x22,.gif) In other words, the x- and y-coordinates of the midpoint are the averages of the x- andy-coordinates of the endpoints. Here’s a practice question: What is the midpoint of the line segment whose endpoints are (6, 0) and (3, 7)? To solve, all you have to do is plug the points given into the midpoint formula, x1 = 6,y1 = 0, x2 = 3, and y2 = 7: (http://img.sparknotes.com/content/testprep/bookimgs/sat2/math2c/0053/midpoint=(6+32.gif) Lines Lines may be nothing more than an infinite set of points arrayed in a straight formation, but there are a number of ways to analyze them. We’ll look at some of the main properties, formulas, and rules of lines. Slope The slope of a line is a measurement of how steeply the line climbs or falls as it moves from left to right. More technically, it is a line’s vertical change divided by its horizontal change, informally known as “the rise over run.” Given two points on a line, call them (x1y1) and (x2y2), the slope of that line can be calculated using the following formula: (http://img.sparknotes.com/content/testprep/bookimgs/sat2/math2c/0056/slope=y2-y1.gif) The variable most often used to represent slope is m. So, for example, the slope of a line that contains the points (–2, –4) and (6, 1) is: (http://img.sparknotes.com/content/testprep/bookimgs/sat2/math2c/0052/m=1-(-4).gif) POSITIVE AND NEGATIVE SLOPES You can easily determine whether the slope of a line is positive or negative just by looking at the line. If a line slopes uphill as you trace it from left to right, the slope is positive. If a line slopes downhill as you trace it from left to right, the…

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