The GMAT Quantitative Section - Geometry

This section reviews some of the terms that you should be familiar with for the Quantitative section. Be aware that the test will probably not ask you for a particular definition; instead, it will ask you to apply the concept to a specific situation. An understanding of the vocabulary involved will help you do this. Here are a few basic terms: ■ A point is a location in a plane. ■ A line is an infinite set of points contained in a straight path. ■ A line segment is part of a line; a segment can be measured. ■ A ray is an infinite set of points that start at an endpoint and continue in a straight path in one direc- tion only. ■ A plane is a two-dimensional flat surface. CHAPTER Geometry 22 357  Angles Two rays with a common endpoint, called a vertex, form an angle. The following figures show the different types of angles: Acute Right The measure is between 0 and 90 degrees. The measure is equal to 90 degrees. Obtuse Straight The measure is between 90 and 180 degrees. The measure is equal to 180 degrees. Here are a few tips to use when determining the measure of the angles. ■ A pair of angles is complementary if the sum of the measures of the angles is 90 degrees. ■ A pair of angles is supplementary if the sum of the measures of the angles is 180 degrees. ■ If two angles have the same measure, then they are congruent. ■ If an angle is bisected, it is divided into two congruent angles. Lines and Angles When two lines intersect, four angles are formed. 1 2 3 4 vertex – GEOMETRY – 358 Vertical angles are the nonadjacent angles formed, or the opposite angles. These angles have the same measure. For example, m ∠ 1 = m ∠ 3 and m ∠ 2 = m ∠ 4. The sum of any two adjacent angles is 180 degrees. For example, m ∠ 1 ϩ m ∠ 2 = 180. The sum of all four of the angles formed is 360 degrees. If the two lines intersect and form four right angles, then the lines are perpendicular. If line m is per- pendicular to line n, it is written m Ќ n. If the two lines are in the same plane and will never intersect, then the lines are parallel. If line l is parallel to line p, it is written l || p. Parallel Lines and Angles Some special angle patterns appear when two parallel lines are cut by another nonparallel line, or a transversal. When this happens, two different-sized angles are created: four angles of one size, and four of another size. ■ Corresponding angles. These are angle pairs 1 and 5, 2 and 6, 3 and 7, and 4 and 8. Within each pair, the angles are congruent to each other. ■ Alternate interior angles. These are angle pairs 3 and 6, and 4 and 5. Within the pair, the angles are congruent to each other. ■ Alternate exterior angles. These are angle pairs 1 and 8, and 2 and 7. Within the pair, the angles are congruent to each other. ■ As in the case of two intersecting lines, the adjacent angles are supplementary and the vertical angles have the same measure.  Polygons A polygon is a simple closed figure whose sides are line segments. The places where the sides meet are called the vertices of the polygon. Polygons are named, or classified, according to the number of sides in the figure. The number of sides also determines the sum of the number of degrees in the interior angles. 12 3 4 5 6 7 8 t l m l ԽԽ m t is the transversal – GEOMETRY – 359 The total number of degrees in the interior angles of a polygon can be determined by drawing the non- intersecting diagonals in the polygon (the dashed lines in the previous figure). Each region formed is a tri- angle; there are always two fewer triangles than the number of sides. Multiply 180 by the number of triangles to find the total degrees in the interior vertex angles. For example, in the pentagon, three triangles are formed. Three times 180 equals 540; therefore, the interior vertex angles of a pentagon is made up of 540 degrees. The formula for this procedure is 180 (n – 2), where n is the number of sides in the polygon. The sum of the measures of the exterior angles of any polygon is 360 degrees. A regular polygon is a polygon with equal sides and equal angle measure. Two polygons are congruent if their corresponding sides and angles are equal (same shape and same size). Two polygons are similar if their corresponding angles are equal and their corresponding sides are in proportion (same shape, but different size).  Triangles Triangles can be classified according to their sides and the measure of their angles. Equilateral Isosceles Scalene All sides are congruent. Two sides are congruent. All sides have a different measure. All angles are congruent. Base angles are congruent. All angles have a different measure. This is a regular polygon. 60° 60° 60° 180° 360° 540° 720° 3-SIDED TRIANGLE 4-SIDED QUADRILATERAL 5-SIDED PENTAGON 6-SIDED HEXAGON – GEOMETRY – 360 Acute Right Obtuse The measure of each It contains one It contains one angle that is greater than 90 angle is less than 90 90-degree angle. degrees. degrees. Triangle Inequality The sum of the two smaller sides of any triangle must be larger than the third side. For example, if the meas- ures 3, 4, and 7 were given, those lengths would not form a triangle because 3 + 4 = 7, and the sum must be greater than the third side. If you know two sides of a triangle and want to find a third, an easy way to han- dle this is to find the sum and difference of the two known sides. So, if the two sides were 3 and 7, the meas- ure of the third side would be between 7 – 3 and 7 + 3. In other words, if x was the third side, x would have to be between 4 and 10, but not including 4 or 10. Right Triangles In a right triangle, the two sides that form the right angle are called the legs of the triangle. The side oppo- site the right angle is called the hypotenuse and is always the longest side of the triangle. Pythagorean Theorem To find the length of a side of a right triangle, the Pythagorean theorem can be used. This theorem states that the sum of the squares of the legs of the right triangle equal the square of the hypotenuse. It can be expressed as the equation a 2 + b 2 = c 2 ,where a and b are the legs and c is the hypotenuse. This relationship is shown geometrically in the following diagram. b 2 2 c 2 a b c a angle greater than 90° 60° 50° 70° – GEOMETRY – 361 Example Find the missing side of the right triangle ABC if the m∠ C = 90°, AC = 6, and AB = 9. Begin by drawing a diagram to match the information given. By drawing a diagram, you can see that the figure is a right triangle, AC is a leg, and AB is the hypotenuse. Use the formula a 2 + b 2 = c 2 by substituting a = 6 and c = 9. a 2 + b 2 = c 2 6 2 + b 2 = 9 2 36 + b 2 = 81 36 – 36 + b 2 = 81 – 36 b 2 = 45 b = ͙ෆ45 which is approximately 6.7 Special Right Triangles Some patterns in right triangles often appear on the Quantitative section. Knowing these patterns can often save you precious time when solving this type of question. 45 — 45 — 90 R IGHT T RIANGLES If the right triangle is isosceles, then the angles’ opposite congruent sides will be equal. In a right triangle, this makes two of the angles 45 degrees and the third, of course, 90 degrees. In this type of triangle, the measure of the hypotenuse is always ͙ ෆ 2 times the length of a side. For example, if the measure of one of the legs is 5, then the measure of the hypotenuse is 5͙ ෆ 2. 5 5 5 2 √ ¯¯¯ 45° 45° A B C 6 9 b – GEOMETRY – 362 30 — 60 — 90 R IGHT T RIANGLES 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 Trapezoid 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 [...]... Strategies for the Quantitative Section 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... the same square (i.e., 22 = 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 is the length of the 2 hypotenuse Know the rule for 30—60—90 right triangles: The shortest side doubled is the hypotenuse and the. .. shortest side times ͙ෆ is the side across from the 60-degree angle 3 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... 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 problems The diagram may or may not conform... 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... that is reasonable 367 – TIPS AND STRATEGIES FOR THE QUANTITATIVE SECTION – ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 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... odd, the result is negative In questions that use a unit of measurement (such as meters, pounds, and so on), be sure that all necessary 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. .. 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... 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 368 . form the right angle are called the legs of the triangle. The side oppo- site the right angle is called the hypotenuse and is always the longest side of the. Pythagorean Theorem To find the length of a side of a right triangle, the Pythagorean theorem can be used. This theorem states that the sum of the squares of the