Arithmatic english 6 ppsx

30 — 60 — 90 RIGHT TRIANGLES In this type of right triangle, a different pattern occurs. Begin with the smallest side of the triangle, which is the side opposite the 30-degree angle. The smallest side multiplied by ͙ ෆ 3 is equal to the side opposite the 60-degree angle. The smallest side doubled is equal to the longest side, which is the hypotenuse. For exam- ple, if the measure of the hypotenuse is 8, then the measure of the smaller leg is 4 and the larger leg is 4͙ ෆ 3 Pythagorean Triples Another pattern that will help with right-triangle questions is Pythagorean triples. These are sets of whole numbers that always satisfy the Pythagorean theorem. Here are some examples those numbers: 3 — 4 — 5 5 — 12 — 13 8 — 15 — 17 7 — 24 — 25 Multiples of these numbers will also work. For example, since 3 2 + 4 2 = 5 2 , then each number doubled (6 — 8 — 10) or each number tripled (9 — 12 — 15) also forms Pythagorean triples.  Quadrilaterals A quadrilateral is a four-sided polygon. You should be familiar with a few special quadrilaterals. Parallelogram This is a quadrilateral where both pairs of opposite sides are parallel. In addition, the opposite sides are equal, the opposite angles are equal, and the diagonals bisect each other. 30° 60° 8 4 √ ¯¯¯ 3 4 – GEOMETRY– 363 Rectangle This is a parallelogram with right angles. In addition, the diagonals are equal in length. Rhombus This is a parallelogram with four equal sides. In addition, the diagonals are perpendicular to each other. Square This is a parallelogram with four right angles and four equal sides. In addition, the diagonals are perpendicular and equal to each other.  Circles ■ Circles are typically named by their center point. This circle is circle C. F G A C B D C 40° E – GEOMETRY– 364 ■ The distance from the center to a point on the circle is called the radius,or r. The radii in this figure are CA, CE, and CB. ■ A line segment that has both endpoints on the circle is called a chord. In the figure, the chords are . ■ A chord that passes through the center is called the diameter,or d. The length of the diameter is twice the length of the radius. The diameter in the previous figure is . ■ A line that passes through the circle at one point only is called a tangent. The tangent here is line FG. ■ A line that passes through the circle in two places is called a secant. The secant in this figure is line CD. ■ A central angle is an angle whose vertex is the center of the circle. In this figure, ∠ACB, ∠ACE, and ∠BCE are all central angles. (Remember, to name an angle using three points, the middle letter must be the vertex of the angle.) ■ The set of points on a circle determined by two given points is called an arc. The measure of an arc is the same as the corresponding central angle. Since the m ∠ACB = 40 in this figure, then the measure of arc AB is 40 degrees. ■ A sector of the circle is the area of the part of the circle bordered by two radii and an arc (this area may resemble a slice of pie). To find the area of a sector, use the formula , where x is the degrees of the central angle of the sector and r is the radius of the circle. For example, in this figure, the area of the sector formed by ∠ACB would be = = = ■ Concentric circles are circles that have the same center.  Measurement and Geometry Here is a list of some of the common formulas used on the GMAT exam: A 4␲ 1 9 × 36␲ 460 360 × ␲6 2 x 360 × ␲6 2 BE BE and CD – GEOMETRY– 365 ■ The perimeter is the distance around an object. Rectangle P = 2l + 2w Square P = 4s ■ The circumference is the distance around a circle. Circle C = ␲d ■ Area refers to the amount of space inside a two-dimensional figure. Parallelogram A = bh Triangle A = ᎏ 1 2 ᎏ bh Tr ap ezoid A = ᎏ 1 2 ᎏ h (b 1 + b 2 ), where b 1 and b 2 are the two parallel bases Circle A = πr 2 ■ The volume is the amount of space inside a three-dimensional figure. General formula V = Bh,where B is the area of the base of the figure and h is the height of the figure Cube V = e 3 ,where e is an edge of the cube Rectangular prism V = lwh Cylinder V = πr 2 h ■ The surface area is the sum of the areas of each face of a three-dimensional figure. Cube SA = 6e 2 ,where e is an edge of the cube Rectangular solid SA = 2(lw) + 2 (lh) + 2(wh) Cylinder SA = 2πr 2 + dh Circle Equations The following is the equation of a circle with a radius of r and center at (h, k): The following is the equation of a circle with a radius of r and center at (0, 0): x 2 ϩ y 2 ϭ r 2 1x Ϫ h2 2 ϩ 1y Ϫ k2 2 ϭ r 2 – GEOMETRY– 366 The following bullets summarize some of the major points discussed in the lessons and highlight critical things to remember while preparing for the Quantitative section. Use these tips to help focus your review as you work through the practice questions. ■ When multiplying or dividing an even number of negatives, the result is positive, but if the number of negatives is odd, the result is negative. ■ In questions that use a unit of measurement (such as meters, pounds, and so on), be sure that all neces- sary conversions have taken place and that your answer also has the correct unit. ■ Memorize frequently used decimal, percent, and fractional equivalents so that you will recognize them quickly on the test. ■ Any number multiplied by zero is equal to zero. ■ A number raised to the zero power is equal to one. ■ Remember that division by zero is undefined. ■ For complicated algebra questions, substitute or plug in numbers to try to find an answer choice that is reasonable. CHAPTER Tips and Strategies for the Quantitative Section 23 367 ■ When given algebraic expressions in fraction form, try to cancel out any common factors in order to simplify the fraction. ■ When multiplying like bases, add the exponents. When dividing like bases, subtract the exponents. ■ Know how to factor the difference between two squares: x 2 – y 2 = (x + y)(x – y). ■ Use FOIL to help multiply and factor polynomials. For example, (x + y) 2 = (x + y)(x + y) = x 2 + xy + xy + y 2 = x 2 + 2xy + y 2 . ■ When squaring a number, two possible choices result in the same square (i.e., 2 2 = 4 and [–2] 2 = 4). ■ Even though the total interior degree measure increases with the number of sides of a polygon, the sum of the exterior angles is always 360 degrees. ■ Know the rule for 45 — 45 — 90 right triangles: The length of a leg multiplied by ͙ ෆ 2 is the length of the hypotenuse. ■ Know the rule for 30 — 60 — 90 right triangles: The shortest side doubled is the hypotenuse and the short- est side times ͙ ෆ 3 is the side across from the 60-degree angle. ■ The incorrect answer choices for problem solving questions will often be the result of making common errors. Be aware of these traps. ■ To solve the data-sufficiency questions, try to solve the problem first using only statement (1). If that works, the correct answer will be either a or d. If statement (1) is not sufficient, the correct answer will be b, c, or e. ■ To save time on the test, memorize the directions and possible answer choices for the data-sufficiency questions. ■ With the data-sufficiency questions, stop as soon as you know if you have enough information. You do not actually have to complete the problem. ■ Although any figures used will be drawn to scale, be wary of any diagrams in data-sufficiency prob- lems. The diagram may or may not conform with statements (1) and (2). ■ Familiarize yourself with the monitor screen and mouse of your test-taking station before beginning the actual exam. Practice basic computer skills by taking the tutorial before the actual test begins. ■ Use the available scrap paper to work out problems. You can also use it as a ruler on the computer screen, if necessary. Remember, no calculators are allowed. ■ The HELP feature will use up time if it is used during the exam. ■ A time icon appears on the screen, so find this before the test starts and use it during the test to help pace yourself. Remember, you have on average about two minutes per question. ■ Since each question must be answered before you can advance to the next question, on problems you are unsure about, try to eliminate impossible answer choices before making an educated guess from the remaining selections. ■ Only confirm an answer selection when you are sure about it — you cannot go back to any previous questions. Reread the question a final time before selecting your answer. ■ Spend a bit more time on the first few questions — by getting these questions correct, you will be given more difficult questions. More difficult questions score more points. – TIPS AND STRATEGIES FOR THE QUANTITATIVE SECTION– 368 . list of some of the common formulas used on the GMAT exam: A 4␲ 1 9 × 36 460 360 × 6 2 x 360 × 6 2 BE BE and CD – GEOMETRY– 365 ■ The perimeter is the distance around an object. Rectangle P =. are equal, the opposite angles are equal, and the diagonals bisect each other. 30° 60 ° 8 4 √ ¯¯¯ 3 4 – GEOMETRY– 363 Rectangle This is a parallelogram with right angles. In addition, the diagonals. with a radius of r and center at (0, 0): x 2 ϩ y 2 ϭ r 2 1x Ϫ h2 2 ϩ 1y Ϫ k2 2 ϭ r 2 – GEOMETRY– 366 The following bullets summarize some of the major points discussed in the lessons and highlight