Chapter 8 Geometry ANGLING FOR A BETTER GRADE Now that you ve reached this chapter you may be having a flashback to your freshman year in high school when you first came in contact with theorems postulates and definitions all woven together to form the geometric proof Well relax GRE geometry questions have little to do with deductive reasoning You re much more likely to be tested on the basic formulas involving area perimeter volume and angle measurements As you work through the GRE Math you ll find that there is a basic battery of terms and formulas that you should know for the geometry questions that do come your way Before we get to those let s look at some techniques Plug In If a problem tells you that a rectangle is x inches long and y inches wide plug in some real numbers to help the question take on a more tangible quality And if there is more than one variable remember to plug in a different number for each one Use Ballparking If a diagram is drawn to scale you can sometimes estimate the right answer and eliminate all the answer choices that don t come close Redraw to Scale If a diagram is not drawn to scale and for problem solving questions you ll know because you ll see the words Note Figure not drawn to scale right below the picture redraw it to make it look like it s supposed to look like Drawings like this are meant to confuse you by suggesting that the figure looks how it s represented in the problem Redrawing the figure to scale helps you avoid falling into that trap Drawn to Scale Problem Solving Problem Solving figures are typically drawn to scale When they are not drawn to scale ETS adds Note Figure not drawn to scale beneath the figure Drawn to Scale Quant Comp Quant Comp figures are often drawn to scale but sometimes they aren t to scale If they aren t ETS does not add any sort of warning like they do for problem solving Check the information in the problem carefully and be suspicious of the figure BASIC HINTS FOR GEOMETRY QUESTIONS Geometry is a special science all its own but that doesn t mean it marches to the beat of an entirely different drummer Many of the techniques you ve learned for other problems will work here as well So let s start with our pal Euclid and his three primary building blocks of measured space points lines and planes LINES AND ANGLES Two points determine a line and two intersecting lines form an angle measured in degrees There are 360 in a complete circle so halfway around the circle forms a straight angle which measures 180 and half of that is a right angle which measures 90 Two lines that intersect in a right angle are perpendicular and perpendicularity is denoted by the symbol Two lines that lie in the same plane and never intersect are parallel which is denoted by Take a look at two parallel lines below If one line intersects two parallel lines it is called a transversal It may look as though this transversal creates eight angles with the two lines but there are actually only two types big angles and small angles All the big angles have the same degree measure and all the small angles have the same degree measure The sum of the degree measures of one big angle and one small angle is always 180 l1 l2 Notice that for the figure above the acute small angles labeled 1 3 5 and 7 are all the same because l1 is parallel with l2 You also know that the angles labeled 2 4 6 and 8 are all the same for the same reason Whenever the GRE states that two lines are parallel look to see if the question is actually testing this concept TRIANGLES Three points determine a triangle and all triangles have three sides and three angles The sum of the measures of the angles inside a triangle is 180 The sides and angles are related Just remember that the longest side is always opposite the largest angle and the shortest side is always opposite the smallest angle Types of Triangles The properties of triangles start to get more interesting when some or all of the sides have the same length Isosceles Triangles If two sides of a triangle have the same length the triangle is isosceles The relationship between sides and angles still goes if two sides of a triangle are the same length then the angles opposite those sides have the same degree measure Equilateral Triangles Equilateral triangles have three equal sides and three equal angles Because the sum of the angle measures is 180 each angle in an equilateral triangle measures or 60 Right Triangles Right triangles contain exactly one right angle and two acute angles The perpendicular sides are called legs and the longest side which is opposite the right angle is called the hypotenuse The Pythagorean Theorem Whenever you know the length of two sides of a right triangle you can find the length of the third side by using the Pythagorean Theorem a2 b2 c2 Most of the time one of the side lengths of a right triangle is irrational and in the form of a square root Any set of three integers that works in the Pythagorean Theorem is called a Pythagorean triple and they re very useful to know for the GRE because they come up often The most common triple is 3 4 5 because 32 42 52 but the other three worth memorizing are 5 12 13 7 24 25 and 8 15 17 All multiples of Pythagorean triples also work in the Pythagorean Theorem If you multiply 3 4 5 by 2 you get 6 8 10 which also works Special Triangles Two specific types of right triangles are called special right triangles because their angles and sides have measurements in a fixed ratio The first an isosceles right triangle is also referred to as a 45 45 90 triangle because its angle measures are 45 45 and 90 The ratio of its side lengths is x x The second is a 30 60 90 triangle which has side lengths in a ratio of x x 2x ETS likes to use special triangles because they can confuse test takers into thinking they don t have enough information to answer a question The fact is though if you know the length of one side of a special triangle you can use the ratios to find the lengths of the other two sides Here s How to Crack It The figure consists of two special triangles consider the 45 45 90 triangle on the left If PN 4 then MN 4 and MP 4 The height PN also helps you find the lengths of the 30 60 90 triangle on the right Because PN is the short side the hypotenuse PO is twice as long or 8 inches long The other side NO measures 4 The perimeter of triangle MPO is therefore 4 4 …