ACT Coordinate Geometry Review Here is a brief review of the coordinate geometry concepts tested on the ACT. Note: there is no review of how to graph an equation on this worksheet. Questions testing this concept are infrequent and if you do see one, remember that you can just plug the equation into your graphing calculator and have it do the work for you! Contents Distance between points... Midpoint of Line Segments... Slope... Linear Equations: Slope-Point Form... 4 Linear Equations: Slope-intercept form... 4 Parallel and Perpendicular Lines... 5 Practice Problems:... 6 1
Distance between points If given the following graph, how could we find the distance between the points? (-4, 6) (-1, -) x y On the ACT, you can usually turn the line into a hypotenuse of a right triangle, find the legs, and use the Pythagorean theorem. You can also use the distance formula equation: Example: ( ) ( ) 1 1 d = x x + y y Find the distance between the following points: Ex. (-4, 6) and (-1, -) (-4, 6) (-1, -) You can think about turning this into a right triangle (see dotted lines) and add an imaginary point at the right angle (see dot connecting the dotted lines). How long would the legs be? The vertical leg goes up the y axis from - all the way to 6, so it would be 8 units long. The horizonta l leg goes across the x axis from -4 to -1, so it is units long. Pythagorean the orm =, so 8 = 7, so c = 7 Or, you can find the distance using the distance formula:
( 1 4) ( 6) d = + + d = + ( ) ( 8) d = 9+ 64 d = 7 Midpoint of Line Segments Given the points: ( x, y ),( x, y ) 1 1 : x + x y + y, m, ( x y ) m ( x y ) 1 1 = Example: Find the midpoints of the following points: (5, 8) and (, -4) 5+ 8 4 m, m =, mp = (4,) Slope The slope of a line measures the change in vertical distance, rise, in relation to the change in horizontal distance, run. Slope rise run vertical = rise horizontal run The formula for the slope between two points (, ) (, ) the change in y and the run is the change in x: y y m = x x x y and x y, given that the rise is 1 1 1 1
Example: Find the slopes of the line segments containing the following pairs of points: (1, -) and (-, 1) m = = Remember: If a line has a positive slopes, it rises from left to right If a line has a negative slope, it falls from left to right. A slope of 0 means that the line is horizontal. An undefined slope means that the line is vertical. Linear Equations: Slope-Point Form The ACT writers like to test the creation of equations. Often you will be given information such as the slope and one point on a line, or just two points from a given line. From that information, you will need to create an equation for those points. It is sometimes helpful to remember the Point-Slope form of an equation which is given by: ( ) y y = m x x 1 1 Where the slope of an equation is m and ( x y ) 1, 1 is any point on the line. Example: The slope of a line is - and the point (4, 1) is on the line. What is the equation of the line? y 1 = -(x 4) y 1 = -x +1 x + y 1 = 0 y = -x + 1 Linear Equations: Slope-intercept form 4
The ACT usually tests equations in y = mx + b form, also called slope-intercept form. Remember that m = slope, b = the y-intercept, and x and y represent any given coordinates on the line. Intercepts are points on the coordinate plane where a graph crosses through the x-axis (called x-intercepts) and the y-axis (called y-intercepts). Example: Find the slope and y-intercept of the line 5x + y = 10. Isolate the y to get the equation into y=mx+b form: y = 5x + 10 5 y = x+ 5 5 m=, b= 5(y-intercept) Parallel and Perpendicular Lines When lines are parallel, their slopes are equal. When lines are perpendicular to each other, their slopes are negative reciprocals of each other. Example: Find the equation of a line parallel to x - y = 6 that passes through the point (, 4) Put equation into form y=mx+b to see the slope: y = x + 6 m = y = x Using the point slope form: y 4= ( x ) ( y 4) = ( x ) y 1= x 6 y = x+ 6 y = x+ Ex. Find the equation of a line perpendicular to the line y x + 4 = 0 that passes through the point (-, -4) 5
Put in y=mx+b form to find slope: y = x 4 1 m = y = x Using the point slope form: y+ 4= ( x+ ) y+ 4= x 4 y = x 8 Practice Problems: 1. What is the slope-intercept form of 8x y 1 = 0 A. y = 8x - 1 B. y = 8x + 1 C. y = -8x + 1 D. y = -8x - 1 E. y = 1-8x F. - G. -16 H. -4 J. K. 4 A. 0 B. 1 C. 5 D. 6 E. 9 F.. What is the x-coordinate of the point in the standard coordinate plane at which the two lines y = x + 4 and y = x + 8 intersect?. What is the distance in the standard coordinate plane between the points (1, 0) and (0, 5)? 4. What is the slope of any line parallel to the line x + 9y = 1 6
G. H. J. K. 9 5. When graphed in the standard coordinate plane, which of the following equations does NOT represent a line? A. y x = B. y = x + 4 C. + y = 10 D. y = E. x = 8 F. - 6. What is the slope of any line parallel to the line 8x + 9y = 1 in the standard (x, y) coordinate plane? G. H. 9 J. 1 K. 17 7. In the standard (x, y) coordinate plane, a line segment has endpoints at (, 6) and (9, ). What are the coordinates of the midpoint of the line segment? A. (, -1) B. (5, -4) C. (6, -4) D. (6, 4) E. (7, ) Answer key: 1. A. H (remember that if the lines intersect, they share the same (x, y) coordinates at that point, so the x and y in the two equations are equal to each other. Since the y s are equal, you can set x + 4 = x + 8 and solve for x).. D 4. H (change into y = mx + b form to determine slope) 5. C (use your graphing calculator to check if you re not sure, but C represents a parabola). 7
6. F (rewrite the equation in y = mx + b form to find m, the slope. Remember that parallel lines have the same slope.) 7. D 8