ACING THE GED EXAM phần 9 potx

– THE GRE QUANTITATIVE SECTION– 191 The area of a sector is found in a similar way to finding the length of an arc. To find the area of a sector, sim- ply multiply the area of a circle, πr 2 , by the fraction ᎏ 36 x 0 ᎏ , again using x as the degree measure of the central angle. Example: Given x = 60 º and r = 8, find the area of the sector: A = ᎏ 3 6 6 0 0 ᎏ ϫ ( ␲ )8 2 A = ᎏ 1 6 ᎏ ϫ 64( ␲ ) A = ᎏ 6 6 4 ᎏ ( ␲ ) A = ᎏ 3 2 2 ᎏ ( ␲ ) Polygons and Parallelograms A polygon is a closed figure with three or more sides. TERMS RELATED TO POLYGONS ■ Ve rt ice s are corner points, also called endpoints, of a polygon. The vertices in the previous polygon are A, B, C, D, E, and F. ■ A diagonal of a polygon is a line segment between two nonadjacent vertices. The two diagonals indi- cated in the previous polygon are line segments BF and AE. ■ A regular (or equilateral) polygon’s sides are all equal. ■ An equiangular polygon’s angles are all equal. FE D C B A r x r o – THE GRE QUANTITATIVE SECTION– 192 ANGLES OF A QUADRILATERAL A quadrilateral is a four-sided polygon. Since a quadrilateral can be divided by a diagonal into two trian- gles, the sum of its angles will equal 180 + 180 = 360 degrees. INTERIOR ANGLES To find the sum of the interior angles of any polygon, use this formula: S = 180(x – 2) where x is the number of polygon sides. Example: Find the sum of the angles in the following polygon. S = (5 – 2) ϫ 180 S = 3 ϫ 180 S = 540 EXTERIOR ANGLES Similar to the exterior angles of a triangle, the sum of the exterior angles of any polygon equals 360 degrees. 5 1 2 3 4 1 2 3 4 m∠1 + m∠2 + m∠3 + m∠4 = 360° SIMILAR POLYGONS If two polygons are similar, their corresponding angles are equal and the ratio of the corresponding sides is in proportion. Example: These two polygons are similar because their angles are equal and the ratio of the corresponding sides is in proportion. Parallelograms A parallelogram is a quadrilateral with two pairs of parallel sides. In this figure, A ෆ B ෆ ʈ C ෆ D ෆ and B ෆ C ෆ ʈ A ෆ D ෆ . A parallelogram has the following characteristics: ■ Opposite sides are equal (AB = CD and BC = AD). ■ Opposite angles are equal (mЄA = mЄC and mЄB = mЄD). ■ Consecutive angles are supplementary (mЄA + mЄB = 180 º ,mЄB + mЄC = 180 º , mЄC + mЄD = 180 º ,mЄD + mЄA = 180 º ). ■ Diagonals bisect each other. SPECIAL TYPES OF PARALLELOGRAMS There are three types of special parallelograms: ■ A rectangle is a parallelogram that has four right angles. D A B C 60° 10 4 6 18 120° 60° 120° 5 2 3 9 – THE GRE QUANTITATIVE SECTION– 193 ■ A rhombus is a parallelogram that has four equal sides. ■ A square is a paralleloram in which all angles are equal to 90 degrees and all sides are equal to each other. DIAGONALS In all parallelograms, diagonals cut each other in two equal halves. ■ In a rectangle, diagonals are the same length. ■ In a rhombus, diagonals intersect to form 90-degree angles. DC AB AC = DB D CB A AB = BC = CD = DA ∠A = m∠B = m∠C = m∠D m D C B A AB = BC = CD = DA D A B C AB = CD BC = AD m∠A = m∠B = m∠C = m∠D – THE GRE QUANTITATIVE SECTION– 194 ■ In a square, diagonals have both the same length and intersect at 90-degree angles. Solid Figures, Perimeter, and Area You will need to know some basic formulas for finding area, perimeter, and volume on the GRE. It is impor- tant that you can recognize the figures by their names and understand when to use which formula. To begin, it is necessary to explain five kinds of measurement: P ERIMETER The perimeter of an object is simply the sum of the lengths of all its sides. 6 7 4 10 Perimeter = 6 + 7 + 4 + 10 = 27 B C A D AC = DB and AC DB B C A D BD AC – THE GRE QUANTITATIVE SECTION– 195 AREA Area is the space inside of the lines defining the shape. You will need to know how to find the area of several geometric shapes and figures. The formulas needed for each are listed here: ■ To find the area of a triangle, use the formula A = ᎏ 1 2 ᎏ bh. ■ To find the area of a circle, use the formula A = ␲ r 2 . ■ To find the area of a parallelogram, use the formula A = bh. ■ To find the area of a rectangle, use the formula A = lw. ■ To find the area of a square, use the formula A = s 2 or A = ᎏ 1 2 ᎏ d 2 . w l h b r h b = Area – THE GRE QUANTITATIVE SECTION– 196 ■ To find the area of a trapezoid, use the formula A = ᎏ 1 2 ᎏ (b 1 + b 2 )h. V OLUME Volume is a measurement of a three-dimensional object such as a cube or a rectangular solid.An easy way to envi- sion volume is to think about filling an object with water. The volume measures how much water can fit inside. ■ To find the volume of a rectangular solid, use the formula V = lwh. ■ To find the volume of a cube, use the formula V = e 3 . e e = edge width length height h b 1 b 2 d s s s s – THE GRE QUANTITATIVE SECTION– 197 ■ To find the volume of a cylinder, use the formula V = ␲ r 2 h. SURFACE AREA The surface area of an object measures the combined area of each of its faces. The total surface area of a rec- tangular solid is double the sum of the area of the three different faces. For a cube, simply multiply the sur- face area of one of its sides by 6. ■ To find the surface area of a rectangular solid, use the formula A = 2(lw ϩ lh ϩ wh). ■ To find the surface area of a cube, use the formula A = 6e 2 . e e = edge V = lwh width length height 4 4 Surface area of front side = 16. Therefore, the surface area of the cube = 16 ϫ 6 = 96. r h – THE GRE QUANTITATIVE SECTION– 198 ■ To find the surface area of a right circular cylinder, use the formula A = 2 ␲ r 2 + 2 ␲ rh. CIRCUMFERENCE Circumference is the measure of the distance around a circle. ■ To find the circumference of a circle, use the formula C = 2 ␲ r. Coordinate Geometry Coordinate geometry is a form of geometrical operations in relation to a coordinate plane. A coordinate plane is a grid of square boxes divided into four quadrants by both a horizontal (x) and vertical (y) axis. These two axes intersect at one coordinate point—(0,0)—the origin. A coordinate pair, also called an ordered pair, is a specific point on the coordinate plane with the first number representing the horizontal placement and sec- ond number representing the vertical. Coordinate points are given in the form of (x,y). GRAPHING ORDERED PAIRS To graph ordered pairs, follow these guidelines: ■ The x-coordinate is listed first in the ordered pair and tells you how many units to move either to the left or to the right. If the x-coordinate is positive, move to the right. If the x-coordinate is negative, move to the left. ■ 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. If the y-coordinate is negative, move down. Example: Graph the following points: (–2,3), (2,3), (3,–2), and (–3,–2). Circumference – THE GRE QUANTITATIVE SECTION– 199 ■ Notice that the graph is broken into four quadrants with one point plotted in each one. Here is a chart to indicate which quadrants contain which ordered pairs, based on their signs: LENGTHS OF HORIZONTAL AND VERTICAL SEGMENTS Two points with the same y-coordinate lie on the same horizontal line, and two points with the same x-coordinate lie on the same vertical line. Find the distance between a horizontal or vertical segment by taking the absolute value of the difference of the two points. Example: Find the length of the line segment AB and the line segment BC. Points Sign of Coordinates Quadrant (2,3) (–2,3) (–3,–2) (3,–2) (+,+) (–,+) (–,–) (+,–) I II III IV II I III IV (−2,3) (2,3) (−3,−2) (3,−2) – THE GRE QUANTITATIVE SECTION– 200 [...]... values On the GRE, these graphs frequently contain differently shaded bars used to represent different elements Therefore, it is important to pay attention to both the size and shading of the graph Money Spent on New Road Work in Millions of Dollars Comparison of Road Work Funds of New York and California 199 0– 199 5 90 80 70 60 50 KEY 40 New York 30 California 20 10 0 199 1 199 2 199 3 199 4 199 5 Year Broken-Line... writing the change in y-coordinates of any two points on the line over the change of the corresponding x-coordinates (This is also known as the rise over the run.) The last step is to simplify the fraction that results Example: Find the slope of a line containing the points (3,2) and (8 ,9) 202 – THE GRE QUANTITATIVE SECTION – (8 ,9) (3,2) Solution: 9 2 ᎏᎏ 8–3 = ᎏ7ᎏ 5 Therefore, the slope of the line... steps: 1 2 3 4 5 6 First, find the mean of the measurements Subtract the mean from each measurement Square each of the differences Sum the square values Divide the sum by n Choose the nonnegative square root of the quotient Example: x 6 7 7 9 15 16 x Ϫ 10 Ϫ4 (x Ϫ 10)2 Ϫ3 Ϫ3 Ϫ1 5 6 16 9 9 1 25 36 96 In the first column, the mean is 10 STANDARD DEVIATION = 96 Ί¯¯¯ = 4 6 When you find the standard deviation of... values Example: Find the average of 9, 4, 7, 6, and 4 9+ 4+7+6+4 ᎏᎏ 5 30 = ᎏ5ᎏ = 6 The denominator is 5 because there are 5 numbers in the set To find the median of a set of numbers, arrange the numbers in ascending order and find the middle value ■ If the set contains an odd number of elements, then simply choose the middle value Example: Find the median of the number set: 1, 5, 3, 7, 2 First, arrange the. .. will be tested on the GRE The central location of a set of numeric values is defined by the value that appears most frequently (the mode), the number that represents the middle value (the median), and/or the average of all the values (the mean) M EAN AND M EDIAN To find the average, or the mean, of a set of numbers, add all the numbers together and divide by the quantity of numbers in the set sum of values... 5, 7 Then, choose the middle value: 3 The answer is 3 ■ If the set contains an even number of elements, simply average the two middle values Example: Find the median of the number set: 1, 5, 3, 7, 2, 8 First, arrange the set in ascending order: 1, 2, 3, 5, 7, 8 Then, choose the middle values 3 and 5 3+5 Find the average of the numbers ᎏ2ᎏ = 4 The answer is 4 M ODE The mode of a set of numbers is the. .. line is ᎏ7ᎏ 5 NOTE: If you know the slope and at least one point on a line, you can find the coordinate point of other points on the line Simply move the required units determined by the slope In the example above, from (8 ,9) , given the slope ᎏ7ᎏ, move up seven units and to the right five 5 units Another point on the line, thus, is (13,16) I MPORTANT I NFORMATION ABOUT S LOPE The following are a few rules... substitute the necessary information into the previous formula based on the following: ■ ■ ■ ■ ■ 100 is always written in the denominator of the percentage-sign column If given a percentage, write it in the numerator position of the number column If you are not given a percentage, then the variable should be placed there The denominator of the number column represents the number that is equal to the whole,... outcomes = ᎏ3ᎏ 5+3+6 3 Therefore, the probability of selecting a red marble is ᎏᎏ 14 2 09 – THE GRE QUANTITATIVE SECTION – M ULTIPLE P ROBABILITIES To find the probability that two or more events will occur, add the probabilities of each For example, in the problem above, if we wanted to find the probability of drawing either a red or blue marble, we would add the probabilities together The probability of... is represented by a chart like the one below The x represents a measurement, and the f represents the number of times that measurement occurs x f total: To use the chart, simply list each measurement only once in the x column and then write how many times it occurs in the f column For example, show the frequency distribution of the following data set that represents the number of students enrolled . and California 199 0– 199 5 New York California KEY 0 10 20 30 40 50 60 70 80 90 199 1 199 2 199 3 199 4 199 5 Money Spent on New Road Work in Millions of Dollars Year 25% 40% 35% – THE GRE QUANTITATIVE. compared, the larger the standard deviation, the larger the dispersion. x 6 7 7 9 15 16 x Ϫ 10 Ϫ4 Ϫ3 Ϫ3 Ϫ1 5 6 (x Ϫ 10) 2 16 9 9 1 25 36 96 STANDARD DEVIATION = Ί ¯¯¯ 96 6 = 4 In the first column, the. ᎏ nu s m um be o r f o v f a v lu al e u s es ᎏ Example: Find the average of 9, 4, 7, 6, and 4. ᎏ 9+ 4+7 5 +6+4 ᎏ = ᎏ 3 5 0 ᎏ = 6 The denominator is 5 because there are 5 numbers in the set. To find the median of a set of numbers, arrange the