Geometry
Coordinate Grids
Every math concept covered so far can be placed on a coordinate grid. Standard two-dimensional coordinate grids are set up with a horizontal x-axis and a vertical y-axis. The place where these lines meet is the origin. At the origin, x = 0 and y = 0. Values of x left of the y-axis are negative, while those to the right are positive. Values of y above the x-axis are positive, while those below it are negative.
Any point can be expressed on the grid using the formula (x,y), so the origin is always (0, 0).
Point D is at (–2, –2), meaning it’s two spaces to the left and then two spaces down. Point C is at (4, 7), so you go four spaces right and then seven spaces up.
Lines appear on grids quite a bit. On a graph, an equation of a line usually takes the form y = mx + b. The variables x and y stand for the (x, y) of any point on the line, while m is the slope of the line and b is the y-intercept.
Slope
The slope of a line, denoted by the variable m, is the change in y-values divided by the change in the x-values of the line. The slope can be determined from any two points on a line. Take points C (4, 7) and B (0, 5), which are both on line AC.
You can use different points on the same line, like A and B, but the slope will still be the same:
Viewed from left to right, slopes are positive if the line is moving upward and negative if the slope heads down. Without knowing the exact value, you know the slope of line DE is negative. Some people like to remember slope as “rise over run.” Starting at point A, if you “rise” one space and then “run” two spaces over, you find yourself at L, another point on the line with a slope of .
The slope of line w is zero, while the slope of line v is undefined. Plug some points into the slope formula to see for yourself. Or just take our word for it. We never lie about slope.
y-Intercept
The b in y = mx + b is called the y-intercept, and it’s the place the line crosses the y-axis. When a line crosses the y-axis, the value for x is zero. Watch what happens when x equals zero:
Understanding all the parts of the linear equation y = mx + b (especially slope) will help you answer many coordinate grid items. But since anything can be placed on a graph, these items will not be limited to finding the slope. Suppose you were given the diagram below and then told H is a point on the line DE:
If H is at (1, –5), how long would line segment DH be?
First, place H on the grid at (1, –5). You can see a right triangle with DH as its hypotenuse. It’s even a 45-45-90 right triangle:
The answer, by the way, is .
Midpoint
To find the midpoint of line segment DH, use the midpoint formula:
There’s nothing wildly exciting about this formula. You just take the x and y values for the two endpoints and find the average for each one.
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