Midpoint x ϭ ᎏ x 1 ϩ 2 x 2 ᎏ ϭ ᎏ Ϫ3 2 ϩ 5 ᎏ ϭ ᎏ 2 2 ᎏ ϭ 1 Midpoint y ϭ ᎏ y 1 ϩ 2 y 2 ᎏ ϭ ᎏ 5 ϩ 2 (Ϫ5) ᎏ ϭ ᎏ 0 2 ᎏ ϭ 0 Therefore, the midpoint of A ෆ B ෆ is (1,0). Slope The slope of a line measures its steepness. Slope is found by calculating the ratio of the change in y-coordinates of any two points on the line, over the change of the corresponding x-coordinates: slope ϭ ᎏ ho v r e i r z t o ic n a t l a c l h c a h n a g n e ge ᎏ ϭ ᎏ x y 2 2 Ϫ Ϫ y x 1 1 ᎏ Example Find the slope of a line containing the points (1,3) and (Ϫ3,Ϫ2). Slope ϭ ᎏ x y 2 2 Ϫ Ϫ y x 1 1 ᎏ ϭ ᎏ 3 1 Ϫ Ϫ ( ( Ϫ Ϫ 2 3 ) ) ᎏ ϭ ᎏ 3 1 ϩ ϩ 2 3 ᎏ ϭ ᎏ 5 4 ᎏ Therefore, the slope of the line is ᎏ 5 4 ᎏ . Practice Question (5,6) (1,3) (1,3) (Ϫ3,Ϫ2) –GEOMETRY REVIEW– 144 What is the slope of the line shown in the figure on the previous page? a. ᎏ 1 2 ᎏ b. ᎏ 3 4 ᎏ c. ᎏ 4 3 ᎏ d. 2 e. 3 Answer b. To find the slope of a line, use the following formula: slope ϭ ᎏ ho v r e i r z t o ic n a t l a c l h c a h n a g n e ge ᎏ ϭ ᎏ x y 2 2 Ϫ Ϫ y x 1 1 ᎏ The two points shown on the line are (1,3) and (5,6). x 1 ϭ 1 x 2 ϭ 5 y 1 ϭ 3 y 2 ϭ 6 Plug in the points into the formula: slope ϭ ᎏ 6 5 Ϫ Ϫ 3 2 ᎏ slope ϭ ᎏ 3 4 ᎏ Using Slope If you know the slope of a line and one point on the line, you can determine other coordinate points on the line. Because slope tells you the ratio of ᎏ ho v r e i r z t o ic n a t l a c l h c a h n a g n e ge ᎏ , you can simply move from the coordinate point you know the required number of units determined by the slope. Example A line has a slope of ᎏ 6 5 ᎏ and passes through point (3,4). What is another point the line passes through? The slope is ᎏ 6 5 ᎏ , so you know there is a vertical change of 6 and a horizontal change of 5. So, starting at point (3,4), add 6 to the y-coordinate and add 5 to the x-coordinate: y: 4 ϩ 6 ϭ 10 x: 3 ϩ 5 ϭ 8 Therefore, another coordinate point is (8,10). If you know the slope of a line and one point on the line, you can also determine a point at a certain coordi- nate, such as the y-intercept (x,0) or the x-intercept (0,y). Example A line has a slope of ᎏ 2 3 ᎏ and passes through point (1,4). What is the y-intercept of the line? Slope ϭ ᎏ x y 2 2 Ϫ Ϫ y x 1 1 ᎏ , so you can plug in the coordinates of the known point (1,4) and the unknown point, the y-intercept (x,0), and set up a ratio with the known slope, ᎏ 2 3 ᎏ , and solve for x: ᎏ y x 2 2 Ϫ Ϫ y x 1 1 ᎏ ϭ ᎏ 2 3 ᎏ ᎏ 0 x Ϫ Ϫ 1 4 ᎏ ϭ ᎏ 2 3 ᎏ –GEOMETRY REVIEW– 145 ᎏ 0 x Ϫ Ϫ 1 4 ᎏ ϭ ᎏ 2 3 ᎏ Find cross products. (Ϫ4)(3) ϭ 2(x Ϫ 1) Ϫ12 ϭ 2x Ϫ 2 Ϫ12 ϩ 2 ϭ 2x Ϫ 2 ϩ 2 Ϫ ᎏ 1 2 0 ᎏ ϭ ᎏ 2 2 x ᎏ Ϫ ᎏ 1 2 0 ᎏ ϭ x Ϫ5 ϭ x Therefore, the x-coordinate of the y-intercept is Ϫ5, so the y-intercept is (Ϫ5,0). Facts about Slope ■ A line that rises to the right has a positive slope. ■ A line that falls to the right has a negative slope. ■ A horizontal line has a slope of 0. slope ϭ 0 negative slope positive slope –GEOMETRY REVIEW– 146 ■ A vertical line does not have a slope at all—it is undefined. ■ Parallel lines have equal slopes. ■ Perpendicular lines have slopes that are negative reciprocals of each other (e.g., 2 and Ϫ ᎏ 1 2 ᎏ ). Practice Question A line has a slope of Ϫ3 and passes through point (6,3). What is the y-intercept of the line? a. (7,0) b. (0,7) c. (7,7) d. (2,0) e. (15,0) slopes are negative reciprocals equal slopes no slope –GEOMETRY REVIEW– 147 Answer a. Slope ϭ ᎏ y x 2 2 Ϫ Ϫ y x 1 1 ᎏ , so you can plug in the coordinates of the known point (6,3) and the unknown point, the y-intercept (x,0), and set up a ratio with the known slope, Ϫ3, and solve for x: ᎏ y x 2 2 Ϫ Ϫ y x 1 1 ᎏ ϭϪ3 ᎏ 0 x Ϫ Ϫ 6 3 ᎏ ϭϪ3 ᎏ x Ϫ Ϫ 3 6 ᎏ ϭϪ3 Simplify. (x Ϫ 6) ᎏ x Ϫ Ϫ 3 6 ᎏ ϭϪ3(x Ϫ 6) Ϫ3 ϭϪ3x ϩ 18 Ϫ3 Ϫ 18 ϭϪ3x ϩ 18 Ϫ18 Ϫ21 ϭϪ3x ᎏ Ϫ Ϫ 2 3 1 ᎏ ϭ ᎏ Ϫ Ϫ 3 3 x ᎏ ᎏ Ϫ Ϫ 2 3 1 ᎏ ϭ x 7 ϭ x Therefore, the x-coordinate of the y-intercept is 7, so the y-intercept is (7,0). –GEOMETRY REVIEW– 148 Translating Words into Numbers To solve word problems, you must be able to translate words into mathematical operations. You must analyze the language of the question and determine what the question is asking you to do. The following list presents phrases commonly found in word problems along with their mathematical equivalents: ■ A number means a variable. Example 17 minus a number equals 4. 17 Ϫ x ϭ 4 ■ Increase means add. Example a number increased by 8 x ϩ 8 CHAPTER Problem Solving This chapter reviews key problem-solving skills and concepts that you need to know for the SAT. Throughout the chapter are sample ques- tions in the style of SAT questions. Each sample SAT question is fol- lowed by an explanation of the correct answer. 8 149 . x: ᎏ y x 2 2 Ϫ Ϫ y x 1 1 ᎏ ϭ ᎏ 2 3 ᎏ ᎏ 0 x Ϫ Ϫ 1 4 ᎏ ϭ ᎏ 2 3 ᎏ –GEOMETRY REVIEW 14 5 ᎏ 0 x Ϫ Ϫ 1 4 ᎏ ϭ ᎏ 2 3 ᎏ Find cross products. (Ϫ4)(3) ϭ 2(x Ϫ 1) 12 ϭ 2x Ϫ 2 12 ϩ 2 ϭ 2x Ϫ 2 ϩ 2 Ϫ ᎏ 1 2 0 ᎏ ϭ ᎏ 2 2 x ᎏ Ϫ ᎏ 1 2 0 ᎏ ϭ. x: ᎏ y x 2 2 Ϫ Ϫ y x 1 1 ᎏ ϭϪ3 ᎏ 0 x Ϫ Ϫ 6 3 ᎏ ϭϪ3 ᎏ x Ϫ Ϫ 3 6 ᎏ ϭϪ3 Simplify. (x Ϫ 6) ᎏ x Ϫ Ϫ 3 6 ᎏ ϭϪ3(x Ϫ 6) Ϫ3 ϭϪ3x ϩ 18 Ϫ3 Ϫ 18 ϭϪ3x ϩ 18 18 Ϫ 21 ϭϪ3x ᎏ Ϫ Ϫ 2 3 1 ᎏ ϭ ᎏ Ϫ Ϫ 3 3 x ᎏ ᎏ Ϫ Ϫ 2 3 1 ᎏ ϭ x 7. ϭ ᎏ ho v r e i r z t o ic n a t l a c l h c a h n a g n e ge ᎏ ϭ ᎏ x y 2 2 Ϫ Ϫ y x 1 1 ᎏ Example Find the slope of a line containing the points (1, 3) and (Ϫ3,Ϫ2). Slope ϭ ᎏ x y 2 2 Ϫ Ϫ y x 1 1 ᎏ ϭ ᎏ 3 1 Ϫ Ϫ ( ( Ϫ Ϫ 2 3 ) ) ᎏ ϭ ᎏ 3 1 ϩ ϩ 2 3 ᎏ ϭ ᎏ 5 4 ᎏ Therefore,