Lines Tutorial for SAT/GRE Test Preparation
|From the complexities of closed figures, let’s simplify things and think only about lines extending in a plane. The two most important facts about arbitrary lines are that (a) when they cross the opposite angles are equal and (b) on one side of a line, there is always 180 degrees. [In fact, you can derive (a) by knowing (b).]Because there are 180o in a straight <, we know
x + z = 180o
But we also know that
y + z = 180o
Therefore x must equal y.
When two lines run so that they never intersect (even if projected an infinite distance in either direction), they are called parallel lines. When two parallel lines are crossed by a third, there is a special correspondence in the angles that are formed.The marked angles in the figure above are equal.
|Okay, let’s put it all together. What can you say about the two blue angles marked in this picture?If it doesn’t come to you right away, review the previous few pages and look for the hints that you need.
Sometimes in a big city roads going in one direction are called avenues, and the roads crossing these are called streets. The avenues and streets are named with consecutive numbers. That way, to identify any spot in the city, you just need to give the intersection between an avenue and a street.”Meet me at 56th Street and 7th Avenue” works because the corner of 56th and 7th is a single point in the city. Furthermore, when you know two points, you also know the distance between them.
“Let’s walk from 56th and 7th to 83rd and 7th.”
How do you know? Because 83 blocks – 56 blocks = 27 blocks.
|In coordinate geometry we do exactly the same thing. (Okay, let’s be clear, it’s the city planners who borrowed from coordinate geometry.) Here’s how they are similar.Label the vertical axis the y-axis. Label the horizontal axis the x-axis. Instead of blocks we’ll talk about units.
Just as in the city, any point in the coordinate plane can be identified by a pair of x, y coordinates, usually presented as a pair like this: (x,y).
The point (3,2) is a unique point in the grid, at the location x = 3 and y = 2.
The x-axis and the y-axis intersect at the origin, the point with coordinates (0,0).
|With coordinate geometry we can do some interesting things. For instance, it is easy to find the distance between two points.Take the points (3,2) and (7,5). To go from (3,2) to (7,5) you need to move 4 units along the x axis, and 3 units up the y-axis.
So how far is that? Well, in terms of city blocks, it would be 4 + 3 = 7 blocks.
But we can also use the Pythagorean theorem to get the shortest distance, which is
42 + 32 = 22 = 52
So the straight line from point (3,2) to point (7,5) is 5 units long.
Between any two points, one unique straight line can be drawn. The slope of that line tells us about its orientation to the grid. Lines that drop downward, from left to right across the grid have a negative slope; lines that are parallel to the x-axis have zero slope; and lines that rise as they move from left to right have positive slope.A very good way to think about slope is as the rise over run, or equivalently, the change in y divided by the change in x.
For any two points, the change in y, as you move from one point to the next, divided by the change in x, is the slope.
|For example, the slope of the line between points (3,2) and (7,5) ism = (5 – 2) / (7 – 3) = 3 / 4
The slope is positive (as we can see from the picture – it rise from left to right), and has a value of .75.
What does slope mean? Slopes can be very informative, especially when we know the units of the two axes. For instance, suppose the y-axis represented distance and the x-axis represented time. Then the slope, y/x, represents the change in distance for a corresponding change in time. That is a quantity we also know as speed – measured in, for example, miles/hour.