Math Review: Geometry OVERVIEW • Lines and angles • Triangles • Isosceles and equilateral triangles • Rectangles, squares, and parallelograms • Circles • Advanced circle problems • Polygons • Cubes and other rectangular solids • Cylinders • Coordinate signs and the four quadrants • Defining a line on the coordinate plane • Graphing a line on the coordinate plane • Midpoint and distance formulas • Coordinate geometry • Summing it up In this chapter, you’ll review the fundamentals involving plane geometry, starting with the following: • Relationships among angles formed by intersecting lines • Characteristics of any triangle • Characteristics of special right triangles • The Pythagorean theorem • Characteristics of squares, rectangles, and parallelograms • Characteristics of circles chapter 11 273 Then, you’ll review the basics of coordinate geometry: • The characteristics of the xy-plane • Defining and plotting points and lines on the plane • Applying the midpoint and distance formulas to problems involving line segments When we’ve finished reviewing the basics, we’ll take a look at the following advanced topics involving plane and coordinate geometry: • Properties of isosceles and equilateral triangles • Properties of trapezoids • Properties of polygons (including those with more than four sides) • Relationships between arcs and other features of circles • Relationships between circles and tangent lines • Relationships created by combining a circle with another geometric figure (such as a triangle or another circle) • Properties of cubes, other rectangular solids, and cylinders • Plotting and defining 2-dimensional figures (triangles, rectangles, and circles) on the xy-plane LINES AND ANGLES Lines and line segments are the basic building blocks for most GMAT geometry problems. A GMAT geometry question might involve nothing more than intersecting lines and the angles they form. To handle the question, just remember four basic rules about angles formed by intersecting lines: Vertical angles (angles across the vertex from each other and formed by the same two lines) are equal in degree measure, or congruent (≅). In other words, they’re the same size. If adjacent angles combine to form a straight line, their degree measures total 180. In fact, a straight line is actually a 180° angle. If two lines are perpendicular (⊥) to each other, they intersect, forming right (90°) angles. The sum of the measures of all angles where two or more lines intersect at the same point is 360° (regardless of how many angles are involved). Note that the symbol (≅) indicates that two geometric features are congruent, meaning that they are identical (the same size, length, degree measure, etc.). The equation AB ≅ CD means that line segment AB is congruent (equal in length) to line segment CD. The two equations ∠A ≅∠B and m∠A 5 m∠B are two different ways of symbolizing the same relationship: that the angle whose vertex is at point A is congruent (equal in degree measure, or size) to the angle whose vertex is at point B. (The letter m symbolizes degree measure.) 274 PART IV: GMAT Quantitative Section www.petersons.com Angles Formed by Intersecting Lines When two or more lines intersect at the same point, they form a “wheel-spoke” pattern with a “hub.” On the GMAT, “wheel-spoke” questions require you to apply one or more of the preceding four rules. 1. x˚ 40˚ y˚ The figure above shows three intersecting lines. What is the value of x 1 y ? (A) 50 (B) 80 (C) 130 (D) 140 (E) 150 The correct answer is (D). The angle vertical to the one indicated as 40° must also measure 40°. That 40° angle, together with the angles whose measures are x° and y°, combine to form a straight (180°) line. In other words, 40 1 x 1 y 5 180. Thus, x 1 y 5 140. A slightly tougher “wheel-spoke” question might focus on overlapping angles and require you to apply rule 1 (about vertical angles) to determine the amount of the overlap. Look at this next “wheel-spoke” figure: A GMAT question about the preceding figure might test your ability to recognize one of the following relationships: Chapter 11: Math Review: Geometry 275 www.petersons.com x 1 y 2 z 5 180 x 1 y exceeds 180 by the amount of the overlap, which equals z, the angle vertical to the overlapping angle. x 1 y 1 v 1 w 5 360 The sum of the measures of all angles, excluding z, is 360°; z is excluded because it is already accounted for by the overlap of x and y. y 2 w 5 zwequals its vertical angle, so y 2 w equals the portion of y vertical to angle z. Parallel Lines and Transversals GMAT problems involving parallel lines also involve at least one transversal, which is a line that intersects each of two (or more) lines. Look at this next figure, in which l 1 i l 2 and l 3 i l 4 : The upper-left “cluster” of angles 1, 2, 3, and 4 matches each of the three other clusters. In other words: • All the odd-numbered angles are congruent (equal in size) to one another. • All the even-numbered angles are congruent (equal in size) to one another. If you know the size of just one angle, you can determine the size of all 16 angles. 276 PART IV: GMAT Quantitative Section www.petersons.com 2. In the figure above, lines P and Q are parallel to each other. If m∠x 5 75°, what is the measure of ∠y? (A) 75° (B) 85° (C) 95° (D) 105° (E) 115° The correct answer is (D). The angle “cluster” where lines P and R intersect corresponds to the cluster where lines Q and R intersect. Thus, ∠x and ∠y are supplementary (their measures add up to 180°). Given that ∠x measures 75°, ∠y must measure 105°. TRIANGLES The triangle (a three-sided polygon) is the test makers’ favorite geometric figure. You’ll need to understand triangles not only to solve “pure” triangle problems but also to solve certain problems involving four-sided figures, three-dimensional figures, and even circles. After a brief review of the properties of any triangle, you’ll focus on right triangles (which include one right, or 90°, angle). Properties of All Triangles Here are four properties that all triangles share: Length of the sides. Each side is shorter than the sum of the lengths of the other two sides. (Otherwise, the triangle would collapse into a line.) Angle measures. The measures of the three angles total 180°. Angles and opposite sides. Comparative angle sizes correspond to the compara- tive lengths of the sides opposite those angles. For example, a triangle’s largest angle is opposite its longest side. (The sides opposite two congruent angles are also congruent.) Be careful not to take this rule too far: The ratio of angle sizes need not be identical to the ratio of lengths of sides. For example, if a certain triangle has angle measures of 30°, 60°, and 90°, the ratio of the angles is 1:2:3. But this doesn’t mean that the ratio of the opposite sides is also 1:2:3. Chapter 11: Math Review: Geometry 277 www.petersons.com Area. The area of any triangle is equal to one-half the product of its base and its height (or “altitude”): Area 5 1 2 3 base 3 height. You can use any side as the base to calculate area. Right Triangles and the Pythagorean Theorem In a right triangle, one angle measures 90° (and, of course, each of the other two angles measures less than 90°). The Pythagorean theorem expresses the relationship among the sides of any right triangle. In the following expression of the theorem, a and b are the two legs (the two shortest sides) that form the right angle, and c is the hypotenuse—the longest side, opposite the right angle: a 2 1 b 2 5 c 2 For any right triangle, if you know the length of two sides, you can determine the length of the third side by applying the theorem. For example: If the two shortest sides (the legs) of a right triangle are 2 and 3 units long, then the length of the triangle’s third side (the hypotenuse) is = 13 units: 2 2 1 3 2 5 13 5 c 2 ; c 5 = 13 If a right triangle’s longest side (hypotenuse) is 10 units long and another side (one of the legs) is 5 units long, then the third side is 5 = 3 units long: a 2 1 5 2 5 10 2 ; a 2 5 75; a 5 = 75 5 = ~25!~3!55 = 3 PYTHAGOREAN TRIPLETS A Pythagorean triplet is a specific ratio among the sides of a triangle that satisfies the Pythagorean theorem. In each of the following triplets, the first two numbers represent the comparative lengths of the two legs, whereas the third—and greatest—number represents the comparative length of the hypotenuse (on the GMAT, the first four appear far more frequently than the last two): 1:1: = 2 1: = 3:2 3:4:5 5:12:13 8:15:17 7:24:25 1 2 1 1 2 5~ = 2! 2 1 2 1~ = 3! 2 5 2 2 3 2 1 4 2 5 5 2 5 2 1 12 2 5 13 2 8 2 1 15 2 5 17 2 7 2 1 24 2 5 25 2 Each triplet above is expressed as a ratio because it represents a proportion among the triangle’s sides. All right triangles with sides having the same proportion, or ratio, have the same shape. For example, a right triangle with sides of 5, 12, and 13 is smaller but exactly the same shape (proportion) as a triangle with sides of 15, 36, and 39. 278 PART IV: GMAT Quantitative Section ALERT! Do not equate altitude (height) with any particular side. Instead, imagine the base on flat ground, and drop a plumb line straight down from the top peak of the triangle to define height or altitude. The only type of triangle in which the altitude equals the length of one side is the right triangle. www.petersons.com 3. Two boats leave the same dock at the same time, one traveling due east at 10 miles per hour and the other due north at 24 miles per hour. How many miles apart are the boats after 3 hours? (A) 68 (B) 72 (C) 78 (D) 98 (E) 110 The correct answer is (C). The distance between the two boats after 3 hours forms the hypotenuse of a triangle in which the legs are the two boats’ respective paths. The ratio of one leg to the other is 10:24, or 5:12. So you know you’re dealing with a 5:12:13 triangle. The slower boat traveled 30 miles (10 mph 3 3 hours). Thirty corresponds to the number 5 in the 5:12:13 ratio, so the multiple is 6 (5 3 6 5 30). 5:12:13 5 30:72:78. Pythagorean Angle Triplets In two (and only two) of the unique triangles identified in the preceding section as Pythagorean side triplets, all degree measures are integers: The corresponding angles opposite the sides of a 1:1: = 2 triangle are 45°, 45°, and 90°. The corresponding angles opposite the sides of a 1: = 3:2 triangle are 30°, 60°, and 90°. If you know that the triangle is a right triangle (one angle measures 90°) and that one of the other angles is 45°, then given the length of any side, you can determine the unknown lengths. For example: • If one leg is 5 units long, then the other leg must also be 5 units long, while the hypotenuse must be 5 = 2 units long. • If the hypotenuse (the longest side) is 10 units long, then each leg must be 5 = 2 units long. Divide hypotenuse by = 2: 10 = 2 5 10 = 2 2 5 5 = 2 TIP To save valuable time on GMAT right-triangle problems, learn to recognize given numbers (lengths of triangle sides) as multiples of Pythagorean triplets. Chapter 11: Math Review: Geometry 279 www.petersons.com Similarly, if you know that the triangle is a right triangle (one angle measures 90°) and that one of the other angles is either 30° or 60°, then given the length of any side you can determine the unknown lengths. For example: • If the shortest leg (opposite the 30° angle) is 3 units long, then the other leg (opposite the 60° angle) must be 3 = 3 units long, and the hypotenuse must be 6 units long (3 3 2). • If the longer leg (opposite the 60° angle) is 4 units long, then the shorter leg (opposite the 30° angle) must be 4 = 3 3 units long (divide by = 3: 4 = 3 5 4 = 3 3 ), while the hypotenuse must be 8 = 3 3 (twice as long as the shorter leg). • If the hypotenuse is 10 units long, then the shorter leg (opposite the 30° angle) must be 5 units long, while the longer leg (opposite the 60° angle) must be 5 = 3 units long (the length of the shorter leg multiplied by = 3). 4. In the figure below, AC is 5 units long, m∠ABD 5 45°, and m∠DAC 5 60°. How many units long is BD? 60˚ 5 45 ˚ A BD C (A) 7 3 (B) 2 = 2 (C) 5 2 (D) 3 = 3 2 (E) 7 2 The correct answer is (C). To find the length of BD, you first need to find AD. Notice that DADC is a 30°-60°- 90° triangle. The ratio among its sides is 1: = 3:2. Given that AC is 5 units long, AD must be 5 2 units long. (The ratio 1:2 is equivalent to the ratio 5 2 :5. Next, notice that DABD is a 45°-45°-90° triangle. The ratio among its sides is 1:1: = 2. You know that AD is 5 2 units long. Thus, BD must also be 5 2 units long. 280 PART IV: GMAT Quantitative Section www.petersons.com ISOSCELES AND EQUILATERAL TRIANGLES Isosceles Triangles An isosceles triangle has the following two special properties: Two of the sides are congruent (equal in length). The two angles opposite the two congruent sides are congruent (equal in size, or degree measure). If you know any two angle measures of a triangle, you can determine whether the triangle is isosceles. 5. In the figure above, BC is 6 units long, m∠A 5 70°, and m∠B 5 40°. How many units long is AB? (A) 5 (B) 6 (D) 7 (C) 8 (E) 9 The correct answer is (B). Since m∠A and m∠B add up to 110°, m∠C 5 70° (70 1 110 5 180), and you know the triangle is isosceles. What’s more, since m∠A 5 m∠C, AB ≅ BC. Given that BC is 6 units long, AB must also be 6 units long. The line bisecting the angle connecting the two congruent sides divides the triangle into two congruent right triangles. So if you know the lengths of all three sides of an isosceles triangle, you can determine the area of the triangle by applying the Pythagorean theorem. 6. Two sides of a triangle are each 8 units long, and the third side is 6 units long. What is the area of the triangle, expressed in square units? (A) 14 (B) 12 = 3 (C) 18 (D) 22 (E) 3 = 55 The correct answer is (E). Bisect the angle connecting the two congruent sides (as in DABC on the following page). The bisecting line is the triangle’s height (h), and the triangle’s base is 6 units long. Chapter 11: Math Review: Geometry 281 www.petersons.com C B A You can determine the triangle’s height (h) by applying the Pythagorean theorem: 3 2 1 h 2 5 8 2 h 2 5 64 2 9 h 2 5 55 h 5 = 55 A triangle’s area is half the product of its base and height. Thus, the area of DABC 5 1 2 ~6! = 55 5 3 = 55 Equilateral Triangles An equilateral triangle has the following three properties: All three sides are congruent (equal in length) The measure of each angle is 60°. Area 5 s 2 = 3 4 (s 5 any side) Any line bisecting one of the 60° angles divides an equilateral triangle into two right triangles with angle measures of 30°, 60°, and 90°; in other words, into two 1: = 3:2 triangles, as shown in the right-hand triangle in the next figure. (Remember that Pythagorean angle triplet?) In the left-hand triangle, if s 5 6, the area of the triangle 5 9 = 3. To confirm this formula, bisect the triangle into two 30°-60°-90° ~1: = 3:2! triangles (as in the right-hand triangle in the preceding figure). The area of this equilateral triangle is 1 2 ~2! = 3, or = 3. The area of each smaller right triangle is = 3 2 . 282 PART IV: GMAT Quantitative Section www.petersons.com . (proportion) as a triangle with sides of 15, 36, and 39. 278 PART IV: GMAT Quantitative Section ALERT! Do not equate altitude (height) with any particular side. Instead, imagine the base on flat ground,. triangle. The slower boat traveled 30 miles (10 mph 3 3 hours). Thirty corresponds to the number 5 in the 5:12:13 ratio, so the multiple is 6 (5 3 6 5 30) . 5:12:13 5 30: 72:78. Pythagorean Angle Triplets In. another. If you know the size of just one angle, you can determine the size of all 16 angles. 276 PART IV: GMAT Quantitative Section www.petersons.com 2. In the figure above, lines P and Q are parallel