Practice Question A rectangle has a perimeter of 42 and two sides of length 10. What is the length of the other two sides? a. 10 b. 11 c. 22 d. 32 e. 52 Answer b. You know that the rectangle has two sides of length 10. You also know that the other two sides of the rectangle are equal because rectangles have two sets of equal sides. Draw a picture to help you better understand: Based on the figure, you know that the perimeter is 10 ϩ 10 ϩ x ϩ x. So set up an equation and solve for x: 10 ϩ 10 ϩ x ϩ x ϭ 42 20 ϩ 2x ϭ 42 20 ϩ 2x Ϫ 20 ϭ 42 Ϫ 20 2x ϭ 22 ᎏ 2 2 x ᎏ ϭ ᎏ 2 2 2 ᎏ x ϭ 11 Therefore, we know that the length of the other two sides of the rectangle is 11. Practice Question The height of a triangular fence is 3 meters less than its base. The base of the fence is 7 meters. What is the area of the fence in square meters? a. 4 b. 10 c. 14 d. 21 e. 28 Answer c. Draw a picture to help you better understand the problem. The triangle has a base of 7 meters. The height is three meters less than the base (7 Ϫ 3 ϭ 4), so the height is 4 meters: 4 7 10 10 x x –GEOMETRY REVIEW– 135 The formula for the area of a triangle is ᎏ 1 2 ᎏ (base)(height): A ϭ ᎏ 1 2 ᎏ bh A ϭ ᎏ 1 2 ᎏ (7)(4) A ϭ ᎏ 1 2 ᎏ (28) A ϭ 14 The area of the triangular wall is 14 square meters. Practice Question A circular cylinder has a radius of 3 and a height of 5. Ms. Stewart wants to build a rectangular solid with a volume as close as possible to the cylinder. Which of the following rectangular solids has dimension closest to that of the circular cylinder? a. 3 ϫ 3 ϫ 5 b. 3 ϫ 5 ϫ 5 c. 2 ϫ 5 ϫ 9 d. 3 ϫ 5 ϫ 9 e. 5 ϫ 5 ϫ 9 Answer d. First determine the approximate volume of the cylinder. The formula for the volume of a cylinder is V ϭ πr 2 h. (Because the question requires only an approximation, use π≈3 to simplify your calculation.) V ϭ πr 2 h V ≈ (3)(3 2 )(5) V ≈ (3)(9)(5) V ≈ (27)(5) V ≈ 135 Now determine the answer choice with dimensions that produce a volume closest to 135: Answer choice a: 3 ϫ 3 ϫ 5 ϭ 9 ϫ 5 ϭ 45 Answer choice b: 3 ϫ 5 ϫ 5 ϭ 15 ϫ 5 ϭ 75 Answer choice c: 2 ϫ 5 ϫ 9 ϭ 10 ϫ 9 ϭ 90 Answer choice d: 3 ϫ 5 ϫ 9 ϭ 15 ϫ 9 ϭ 135 Answer choice e: 5 ϫ 5 ϫ 9 ϭ 25 ϫ 9 ϭ 225 Answer choice d equals 135, which is the same as the approximate volume of the cylinder. –GEOMETRY REVIEW– 136 Practice Question Mr. Suarez painted a circle with a radius of 6. Ms. Stone painted a circle with a radius of 12. How much greater is the circumference of Ms. Stone’s circle than Mr. Suarez’s circle? a. 3π b. 6π c. 12π d. 108π e. 216π Answer c. You must determine the circumferences of the two circles and then subtract. The formula for the circum- ference of a circle is C ϭ 2πr. Mr. Suarez’s circle has a radius of 6: C ϭ 2πr C ϭ 2π(6) C ϭ 12π Ms. Stone’s circle has a radius of 12: C ϭ 2πr C ϭ 2π(12) C ϭ 24π Now subtract: 24π Ϫ 12π ϭ 12π The circumference of Ms. Stone’s circle is 12π greater than Mr. Suarez’s circle.  Coordinate Geometry A coordinate plane is a grid divided into four quadrants by both a horizontal x-axis and a vertical y-axis. Coor- dinate points can be located on the grid using ordered pairs. Ordered pairs are given in the form of (x,y). The x represents the location of the point on the horizontal x-axis, and the y represents the location of the point on the vertical y-axis. The x-axis and y-axis intersect at the origin, which is coordinate point (0,0). Graphing Ordered Pairs The x-coordinate is listed first in the ordered pair, and it tells you how many units to move to either the left or the right. If the x-coordinate is positive, move from the origin to the right. If the x-coordinate is negative, move from the origin to the left. –GEOMETRY REVIEW– 137 The y-coordinate is listed second and tells you how many units to move up or down. If the y-coordinate is positive, move up from the origin. If the y-coordinate is negative, move down from the origin. Example Graph the following points: (0,0) (3,5) (3,Ϫ5) (Ϫ3,5) (Ϫ3,Ϫ5) Notice that the graph is broken up into four quadrants with one point plotted in each one. The chart below indicates which quadrants contain which ordered pairs based on their signs: POINT SIGNS OF COORDINATES QUADRANT (3,5) (+,+) I (–3,5) (–,+) II (–3,–5) (–,–) III (3,–5) (+,–) IV (Ϫ3,5) (0,0) Quadrant II Quadrant I Quadrant III Quadrant IV (3,5) (Ϫ3,Ϫ5) (3,Ϫ5) –GEOMETRY REVIEW– 138 Practice Question Which of the five points on the graph above has coordinates (x,y) such that x ϩ y ϭ 1? a. A b. B c. C d. D e. E Answer d. You must determine the coordinates of each point and then add them: A (2,Ϫ4): 2 ϩ (Ϫ4) ϭϪ2 B (Ϫ1,1): Ϫ1 ϩ 1 ϭ 0 C (Ϫ2,Ϫ4): Ϫ2 ϩ (Ϫ4) ϭϪ6 D (3,Ϫ2): 3 ϩ (Ϫ2) ϭ 1 E (4,3): 4 ϩ 3 ϭ 7 Point D is the point with coordinates (x,y) such that x ϩ y ϭ 1. Lengths of Horizontal and Vertical Segments The length of a horizontal or a vertical segment on the coordinate plane can be found by taking the absolute value of the difference between the two coordinates, which are different for the two points. A E B D 1 C 1 –GEOMETRY REVIEW– 139 Example Find the length of A ෆ B ෆ and B ෆ C ෆ . A ෆ B ෆ is parallel to the y-axis, so subtract the absolute value of the y-coordinates of its endpoints to find its length: A ෆ B ෆ ϭ |3 Ϫ (Ϫ2)| A ෆ B ෆ ϭ |3 ϩ 2| A ෆ B ෆ ϭ |5| A ෆ B ෆ ϭ 5 B ෆ C ෆ is parallel to the x-axis, so subtract the absolute value of the x-coordinates of its endpoints to find its length: B ෆ C ෆ ϭ |Ϫ3 Ϫ 3| B ෆ C ෆ ϭ |Ϫ6| B ෆ C ෆ ϭ 6 Practice Question A B C (Ϫ2,7) (Ϫ2,Ϫ6) (5,Ϫ6) A B C (Ϫ3,3) (Ϫ3,Ϫ2) (3,Ϫ2) –GEOMETRY REVIEW– 140 What is the sum of the length of A ෆ B ෆ and the length of B ෆ C ෆ ? a. 6 b. 7 c. 13 d. 16 e. 20 Answer e. A ෆ B ෆ is parallel to the y-axis, so subtract the absolute value of the y-coordinates of its endpoints to find its length: A ෆ B ෆ ϭ |7 Ϫ (Ϫ6)| A ෆ B ෆ ϭ |7 ϩ 6| A ෆ B ෆ ϭ |13| A ෆ B ෆ ϭ 13 B ෆ C ෆ is parallel to the x-axis, so subtract the absolute value of the x-coordinates of its endpoints to find its length: B ෆ C ෆ ϭ |5 Ϫ (Ϫ2)| B ෆ C ෆ ϭ |5 ϩ 2| B ෆ C ෆ ϭ |7| B ෆ C ෆ ϭ 7 Now add the two lengths: 7 ϩ 13 ϭ 20. Distance between Coordinate Points To find the distance between two points, use this variation of the Pythagorean theorem: d ϭ ͙(x 2 Ϫ x ෆ 1 ) 2 ϩ ( ෆ y 2 Ϫ y 1 ෆ ) 2 ෆ Example Find the distance between points (2,Ϫ4) and (Ϫ3,Ϫ4). C (2,4) (Ϫ3,Ϫ4) (5,Ϫ6) –GEOMETRY REVIEW– 141 The two points in this problem are (2,Ϫ4) and (Ϫ3,Ϫ4). x 1 ϭ 2 x 2 ϭϪ3 y 1 ϭϪ4 y 2 ϭϪ4 Plug in the points into the formula: d ϭ ͙(x 2 Ϫ x ෆ 1 ) 2 ϩ ( ෆ y 2 Ϫ y 1 ෆ ) 2 ෆ d ϭ ͙(Ϫ3 Ϫ ෆ 2) 2 ϩ ෆ (Ϫ4 Ϫ ෆ (Ϫ4)) ෆ 2 ෆ d ϭ ͙(Ϫ3 Ϫ ෆ 2) 2 ϩ ෆ (Ϫ4 ϩ ෆ 4) 2 ෆ d ϭ ͙(Ϫ5) 2 ෆ ϩ (0) 2 ෆ d ϭ ͙25 ෆ d ϭ 5 The distance is 5. Practice Question What is the distance between the two points shown in the figure above? a. ͙20 ෆ b. 6 c. 10 d. 2͙34 ෆ e. 4͙34 ෆ (1,Ϫ4) (Ϫ5,6) –GEOMETRY REVIEW– 142 Answer d. To find the distance between two points, use the following formula: d ϭ ͙(x 2 Ϫ x ෆ 1 ) 2 ϩ ( ෆ y 2 Ϫ y 1 ෆ ) 2 ෆ The two points in this problem are (Ϫ5,6) and (1,Ϫ4). x 1 ϭϪ5 x 2 ϭ 1 y 1 ϭ 6 y 2 ϭϪ4 Plug the points into the formula: d ϭ ͙(x 2 Ϫ x ෆ 1 ) 2 ϩ ( ෆ y 2 Ϫ y 1 ෆ ) 2 ෆ d ϭ ͙(1 Ϫ (Ϫ ෆ 5)) 2 ϩ ෆ (Ϫ4 Ϫ ෆ 6) 2 ෆ d ϭ ͙(1 ϩ 5 ෆ ) 2 ϩ (Ϫ ෆ 10) 2 ෆ d ϭ ͙(6) 2 ϩ ෆ (Ϫ10) ෆ 2 ෆ d ϭ ͙36 ϩ 1 ෆ 00 ෆ d ϭ ͙136 ෆ d ϭ ͙4 ϫ 34 ෆ d ϭ ͙34 ෆ The distance is 2͙34 ෆ . Midpoint A midpoint is the point at the exact middle of a line segment. To find the midpoint of a segment on the coordi- nate plane, use the following formulas: Midpoint x ϭ ᎏ x 1 ϩ 2 x 2 ᎏ Midpoint y ϭ ᎏ y 1 ϩ 2 y 2 ᎏ Example Find the midpoint of A ෆ B ෆ . B A Midpoint (5,Ϫ5) (Ϫ3,5) –GEOMETRY REVIEW– 143 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 [...]... 6,400 is 16 ■ finding what percentage one number is of another number Example What percentage of 90 is 18? First translate the problem into math: What precentage of 90 is 18? x 100 ϫ 90 ϭ 18 Now solve: x ᎏᎏ ϫ 90 ϭ 18 100 90x ᎏᎏ ϭ 18 100 9x ᎏᎏ ϭ 18 10 9x ᎏᎏ ϫ 10 ϭ 18 ϫ 10 10 9x ϭ 180 x ϭ 20 Answer: 18 is 20% of 90 154 ... Simplify Ϫ3 ϭ Ϫ3x ϩ 18 Ϫ3 Ϫ 18 ϭ Ϫ3x ϩ 18 Ϫ 18 Ϫ21 ϭ Ϫ3x Ϫ21 Ϫ3x ᎏᎏ ϭ ᎏᎏ Ϫ3 Ϫ3 Ϫ21 ᎏᎏ ϭ x Ϫ3 7ϭx Therefore, the x-coordinate of the y-intercept is 7, so the y-intercept is (7,0) 1 48 C H A P T E R 8 Problem Solving This chapter reviews key problem-solving skills and concepts that you need to know for the SAT Throughout the chapter are sample questions in the style of SAT questions Each sample SAT question is... means to multiply English: 10% of 30 equals 3 10 Math: ᎏ0ᎏ ϫ 30 ϭ 3 1 0 ■ The word what refers to a variable English: 20% of what equals 8? 20 Math: ᎏ0ᎏ ϫ a ϭ 8 1 0 ■ The words is, are, and were, mean equals English: 0.5% of 18 is 0.09 0.05 Math: ᎏ0ᎏ ϫ 18 ϭ 0.09 1 0 ■ 0.3 0.3% ϭ ᎏ0ᎏ ϭ 0.003 1 0 When answering a percentage problem, rewrite the problem as math using the translations above and then solve... percentage of a given number Example What number is 80 % of 40? First translate the problem into math: What number is 80 % of 40? x ϭ 80 ϫ 40 100 Now solve: 80 x ϭ ᎏ0ᎏ ϫ 40 1 0 3,200 x ϭ ᎏ0ᎏ 1 0 x ϭ 32 Answer: 32 is 80 % of 40 ■ finding a number that is a percentage of another number Example 25% of what number is 16? First translate the problem into math: 153 – PROBLEM SOLVING – 0.25% of what number is... 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 8 149 – PROBLEM SOLVING... know there is a vertical change of 6 and a horizontal change of 5 So, starting at point 5 (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 coordinate, such as the y-intercept (x,0) or the x-intercept (0,y) Example... equals 4 17 Ϫ x ϭ 4 ■ Increase means add Example a number increased by 8 x 8 149 – PROBLEM SOLVING – ■ More than means add Example 4 more than a number 4ϩx ■ Less than means subtract Example 8 less than a number x 8 ■ Times means multiply Example 6 times a number 6x ■ Times the sum means to multiply a number by a quantity Example 7 times the sum of a number and 2 7(x ϩ 2) ■ Note that variables can be... on the SAT: 1 finding the percentage of a given number Example: What number is 60% of 24? 2 finding a number when a percentage is given Example: 30% of what number is 15? 3 finding what percentage one number is of another number Example: What percentage of 45 is 5? 152 – PROBLEM SOLVING – To answer percent questions, write them as fraction problems To do this, you must translate the questions into math. .. find the possible values of y? a (y ϩ 23)2 ϭ 5y Ϫ 4 b y2 ϩ 23 ϭ 5y Ϫ 4 c y2 ϩ (23)2 ϭ y(4 Ϫ 5) d y2 ϩ (23)2 ϭ 5y Ϫ 4 e (y ϩ 23)2 ϭ y(4 Ϫ 5) Answer a Break the problem into pieces while translating into mathematics: squaring translates to raise something to a power of 2 the sum of y and 23 translates to (y ϩ 23) So, squaring the sum of y and 23 translates to (y ϩ 23)2 gives a result translates to ϭ 4 . percentage of 90 is 18? First translate the problem into math: Now solve: ᎏ 10 x 0 ᎏ ϫ 90 ϭ 18 ᎏ 1 9 0 0 0 x ᎏ ϭ 18 ᎏ 1 9 0 x ᎏ ϭ 18 ᎏ 1 9 0 x ᎏ ϫ 10 ϭ 18 ϫ 10 9x ϭ 180 x ϭ 20 Answer: 18 is 20% of 90. What. 3. Math: ᎏ 1 1 0 0 0 ᎏ ϫ 30 ϭ 3 ■ The word what refers to a variable. English: 20% of what equals 8? Math: ᎏ 1 2 0 0 0 ᎏ ϫ a ϭ 8 ■ The words is, are, and were, mean equals. English: 0.5% of 18. 18 Ϫ3 Ϫ 18 ϭϪ3x ϩ 18 Ϫ 18 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– 1 48  Translating