THE SQUARE OF THE LENGTH OF THE HYPOTENUSE OF A RIGHT TRIANGLE ISE...

180.The square of the length of the hypotenuse of a right triangle isequal to the sum of the squares of the lengths of the two legs(Pythagorean Theorem).The sides of a 45˚– 45˚– 90˚ triangle are in the ratio 1: 1: 2,and the sides of a 30˚– 60˚– 90˚ triangle are in the ratio 1 :3 : 2.Drawing in lines that are not shown in a figure can sometimes behelpful in solving a geometry problem; for example, by drawing thedashed lines in the pentagon below,The x-axis and y-axis intersect at the origin O, and they partitionthe plane into four quadrants, as shown. Each point in the plane hascoordinates (x, y) that give its location with respect to the axes; forthe total number of degrees in the angles of the pentagon can beexample, the point P(2, –8) is located 2 units to the right of the y-found by adding the number of degrees in each of the threeaxis and 8 units below the x-axis. The units on the x-axis are thetriangles: 180 + 180 + 180 = 540.same length as the units on the y-axis, unless otherwise noted.Equations involving the variables x and y can be graphed incompare the quantities that are given in two columns, Column AThe number of degrees of arc in a circle is 360.If O is the center of the circle in the figure below, then the lengthand Column B, and decide whether one quantity is greater than theother, whether the two quantities are equal, or whether the relation-of arc ABC is xship cannot be determined from the information given. Information360 times the circumference of the circle.about the two quantities is given in the columns themselves or may

A

be centered above the columns. Here are some examples with the

B

x°

O

C

correct answers indicated according to the following answerchoices.(A) The quantity in Column A is greater.The volume of a rectangular solid or a right circular cylinder is(B) The quantity in Column B is greater.the product of the area of the base and the height; for example, the(C) The two quantities are equal.volume of a cylinder with a base of radius 2 and a height of 5 is(D) The relationship cannot be determined from(2

²

) (5) = 20the information given.DATA ANALYSISColumn A Column B Correct AnswerQuestions that test data analysis include those involving theExample 1: 2

³

3

²

^A

^B

^C

^D

^E

following topics: basic descriptive statistics (such as mean,median, mode, range, standard deviation, and percentiles),Example 2: The smallest 23

^A

^B

^C

^D

^E

interpretation of data given in graphs and tables (such as barprime numberand circle graphs, and frequency distributions), and elementarygreater than 20probability. The questions assess the ability to synthesize infor-mation, to select appropriate data for answering a question, andm is an integer.to determine whether or not the data provided are sufficient toanswer a given question. The emphasis in these questions is onExample 3: 3m + 7 7

^A

^B

^C

^D

^E

the understanding of basic principles (for example, basic prop-(since m can beerties of normal distribution) and reasoning within the contextpositive, negative,of given information.or zero)Some questions only require some manipulation to determineSome facts about descriptive statistics and probability thatwhich of the quantities is greater; other questions require more rea-may be helpfulsoning or thinking of special cases in which the relative sizes of theIn a distribution of n measurements, the (arithmetic) mean is thequantities are reversed.sum of the measurements divided by n. The median is the middleThe following strategies may help in answering quantitative com-measurement after the measurements are ordered by size if n isparison questions.odd, or it is the mean of the two middle measurements if n is even.Do not waste time performing needless computations in order toThe mode is the most frequently occurring measurement (thereeventually compare two specific numbers. Simplify or transformmay be more than one mode). The range is the difference betweenone or both of the given quantities only as much as is necessarythe greatest measurement and the least measurement. Thus, forto determine which quantity is greater or whether the two quanti-the measurements: 70, 72, 72, 76, 78, and 82, the mean isties are equal. If you determine that one quantity is greater than450675, the median is (7276)274, the mode is 72,the other, do not take time to find the exact sizes of the quantities.and the range is 12.Answer and go on to the next question.The probability that an event will occur is a value between 0Consider all kinds of appropriate real numbers before you makeand 1, inclusive. If p is the probability that a particular event willoccur, then 0 ≤ p≤1, and the probability that the event will not a decision. As soon as you establish that the quantity in oneoccur is 1p. For example, if the probability is 0.85 that it willcolumn is greater in one case while the quantity in the otherrain tomorrow, then the probability that it will not rain tomorrow iscolumn is greater in another case, choose “The relationship10.850.15.cannot be determined from the information given” and moveon to the next question.The quantitative measure employs two types of questions: quan-Geometric figures may not be drawn to scale. Comparisonstitative comparison and problem solving.should be based on the given information together with yourQUANTITATIVE COMPARISONknowledge of mathematics rather than on the exact appearanceof the figure. You can sometimes find a clue by sketching an-The quantitative comparison questions test the ability to reasonother figure that conforms to the information given. (Scratchquickly and accurately about the relative sizes of two quantities orpaper will be provided.) Try to visualize the parts of the figureto perceive that not enough information is provided to make such athat are fixed by the information given and the parts that arecomparison. To solve a quantitative comparison problem, you mustchangeable. If the figure can be changed in such a way thatthe relative sizes of the quantities in the columns are reversedwhile still conforming to the information given, then theanswer is “The relationship cannot be determined from theinformation given.”Column A Column BHere are some more examples: