SAT 8

CHAPTER 6

COORDINATE GEOMETRY

This chapter provides a review of coordinate geometry principles. On the Level 2 test, 10–14% of the questions relate specifically to coordinate geometry. Coordinate geometry focuses on graphing in an xy-coordinate plane. The pie chart shows approximately how much of the Level 2 test is related to coordinate geometry:

Numbers and

Operations

12% Algebra

20%

Data Analysis,

Statistics, and

Probability

8%

Solid

Geometry

4%

Coordinate

Geometry

12%

Functions

30%

Trigonometry

14%

The following geometry topics are covered in this chapter:

1.Plotting Points

2.Midpoint

3.Distance

4.Slope

5.Slope of Parallel and Perpendicular Lines

6.Equations of Lines

a.Horizontal and Vertical Lines

b.Standard Form

c.Point-Slope Form

d.Slope-Intercept Form

e.Determining x and y Intercepts

7.Circles

8.Parabolas

96

9.*Ellipses

10.*Hyperbolas

11.Graphing Inequalities

12.Graphing Absolute Value

13.Symmetry

14.Transformations

15.*Polar Coordinates

PLOTTING POINTS

A plane rectangular coordinate system is created by drawing two axes that intersect at right angles at an origin. The horizontal axis is called the x-axis and the vertical axis is called the y-axis. The axes create 4 quadrants in the plane, which are numbered using Roman numerals as shown:

y

x

III IV

An ordered pair (x, y) denotes a unique point in the plane. To plot a point, simply graph the location of its x and y coordinates. The x-coordinate of a point is also referred to as the abscissa, while the y-coordinate is also referred to as the ordinate.

E X A M P L E :

Graph the following points in the same coordinate plane: A(1, 4), B(−2, −3), C(−5, 1), D(4, 0) (see graph on page 98).

*Denotes concepts that are on the Level 2 test only.

100 PART II / MATH REVIEW

E X A M P L E :

Find the perimeter of ABC given its vertices are A(2, 2), B(−1, 5), and C(−5, 2).

This problem requires applying the distance formula three times to find the length of the three sides of the triangle. Solve for AB, BC, and AC

___

Notice that AC is a horizontal line segment, so the distance from A to C can be easily found be determining the change in the x coordinates.2 − −5 = 7. The perimeter of the triangle is:

The concept of distance can also be applied to finding the area of figures in a coordinate plane. Use the distance formula to find the length of the base and/or height of a figure, and substitute these values into the appropriate area formula. (See Chapter 5 for a review of area formulas.)

E X A M P L E :

Find the area of parallelogram ABCD given its vertices are A(3, 1), B(2, −1), C(−1, −1), and D(0, 1).

Sketch a diagram to help picture the parallelogram.

y

E X A M P L E :

Find the slope of the line containing the points S(6, −9) and R(−2, −5).

m = y2 − y1 . x2 − x1

m = −5 − −9 . −2 − 6

In the above example, we subtracted the coordinates of S from R. Order doesn’t matter, however, as long as you choose the same order in the numerator and the denominator. We could have found the slope by subtracting the coordinates of R from S to get the same answer:

SLOPE OF PARALLEL AND PERPENDICULAR LINES

You can determine whether or not lines are parallel (never intersect) or perpendicular (intersect at a right angle) by examining their slope:

1.Parallel lines have the same slope. m1 = m2.

2.Perpendicular lines have slopes that are opposite reciprocals. m1 m2 = −1.

E X A M P L E :

___

Find the slope of the perpendicular bisector of AB given A(0, −2) and

B(3, 3).

___

The slope of AB is:

m = y2 − y1 . x2 − x1

m = 3 − −2 = 5 . 3 − 0 3

___

The slope of any line perpendicular to AB, including the perpendicular

5 bisector, is the opposite reciprocal of 3 .

EQUATIONS OF LINES

Linear equations are equations whose graphs are straight lines. Equations containing two variables x and y raised to the first power are linear. By definition, linear equations have a constant slope.

The three forms of the equation of a line are further explained in the succeeding paragraphs.

Horizontal and Vertical Lines

The most basic forms of linear equations are horizontal and vertical lines. Horizontal lines are written in the form y = a, where a is any constant. Vertical lines are written in the form x = a, where, again, a is any constant.

E X A M P L E :

Find the point of intersection of the lines y = 7 and x = 5. The graph of the two lines clearly shows their intersection.

(5, 7). (Answer)

Standard Form

The standard form of the equation of a line is:

Ax + By = C (where A and B are both ≠ 0).

The slope of a line in standard form is − BA (B ≠ 0).