# SAT Tip: Getting the Hang of Coordinate Geometry

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This tip on improving your SAT score was provided by Vivian Kerr at Veritas Prep.
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It can feel as if there are a lot of formulas and concepts tested on the SAT, but Coordinate Geometry is pretty predictable. We have to know the standard equations of lines, parabolas, and circles graphed on a coordinate plane, and we have to know how to find the midpoint and distance between points on those shapes.

**Distance Formula** =

We use this to find the distance between any two points (x_{1}, y_{1}) and (x_{2}, y_{2}). Note how it’s basically a derivative of the Pythagorean Theorem (imagine drawing a right triangle so that the distance between these points is the hypotenuse).

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**Midpoint Formula** =

Use this to find the midpoint between two points (notice how you are essentially finding the average of the x-coordinates and the average of the y-coordinates). Another big concept when we’re talking about lines is slope. The slope is described as Rise / Run or Change in y / Change in x. As long as you know any two points on a line, you can find the slope. Remember that parallel lines have the *same* slope, and perpendicular lines have *negative reciprocal* slopes.

**y = mx + b **This is called slope-intercept form. An equation in this form will always make a straight line on a graph (note how neither x nor y have an exponent). In this form, b is the y-intercept (the point on the y-axis where the line crosses) and m is the slope.

**y = ax ^{2} + bx + c **This is the standard equation for a parabola. In this equation, c represents the y-intercept. A standard equation in which a variable is squared will

*never*make a straight line.

**(x – h) ^{2} + (y – k)^{2} = r^{2 }**This is the standard equation for a circle. Here

*(h, k)*is the center point of the circle and

*r*is the radius. Notice that inside the parentheses, we are subtracting “h” and “k” when on the graph, “h” and “k” are both positive. If “h” and “k” were negative, we would add them inside the parentheses. On test day, you may be given parameters and asked the question:

* 1. What is the equation of a circle with a center at (4, -5) and a diameter of 16?*

To solve, let’s start with our standard equation:** **(x – h)^{2} + (y – k)^{2} = r^{2 }

The radius is half the diameter in a circle, so the radius would be 8. 8^{2} = 64. We’re going to take the opposite sign inside the parentheses (since “4” is positive we will subtract it, and since “-5” is negative we will add it). The correct answer is (x – 4)^{2} + (y + 5)^{2} = 64.

Remember that any straight line can be described by the equation y = mx + b. On test day, as long as you have two coordinates of a line, you can always find its equation.

For Coordinate Geometry question practice, take a full-length SAT practice test to sharpen your skills.