3 Ways to Solve a 750+ Level GMAT Question About Irregular Polygons

We have examined how to deal with polygons when you encounter them on a GMAT question in a previous post.  Today, we will look at a relatively difficult polygon question, however we would like to remind you here that the concepts being tested in this question are still very simple (although we wont give away exactly which concepts they are yet). First, take a look at the question itself: The hexagon above has interior angles whose measures are all equal. As shown, only five of the six side lengths are known: 10, 15, 4, 18, and 7. What is the unknown side length? (A)  7 (B)10 (C) 12 (D) 15 (E) 16 There are various ways to solve this question, but each takes a bit of effort. Note that the polygon we are given is not a regular polygon, since the side lengths are not all equal. The angles, however,  are all equal. Let’s first find the measure of each one of those angles using the formula discussed in  this previous post. (n 2)*180 = sum of all interior angles (6 2)*180 = 720 Each of the 6 angles = 720/6 = 120 degrees Though we would like to point out here that if you see a question such as this one on the actual GMAT exam, you should already know that if each angle of a hexagon is equal, each angle must be 120 degrees, so performing the above calculation would not be necessary. Method 1: Visualization This is a very valid approach to obtaining the correct answer on this  GMAT question since we don’t need to explain the reasoning or show our  steps, however  it may be hard to comprehend for the beginners. We will try to explain it anyway, since it requires virtually  no work and will help build your math instinct. Note that in  the given hexagon, each angle is 120 degrees this means that each pair of opposite sides are parallel. Think of it this way: Side 4 turns on Side 18 by 120 degrees. Then Side 15 turns on Side 4 by another 120 degrees. And finally, Side 10 turns on Side 15 by another 120 degrees. So Side 10 has, in effect, turned by 360 degrees on Side 18. This means Side 10 is parallel to Side 18. Now, think of the 120 degree angle between Side 4 and Side 15   it has to be kept constant. Plus, the angles of the legs must also stay constant at 120 degrees with Sides 10 and 18. Since the slopes of each leg of that angle  are negatives of each other (√3 and -√3), when one leg gets shorter, the other gets longer by the same length (use the image below as a visual of what were talking about). Hence, the sum of the sides will always be 15 + 4 = 19. This means 7 + Unknown = 19, so  Unknown = 12. Our answer is C. If you struggled to understand the approach above, youre not alone. This method involves a lot of intuition, and struggling to figure it out may not be the best use of your time on the GMAT, so let’s examine  a couple of more tangible solutions! Method 2: Using Right Triangles As we saw  in Method 1 above, AB and DE are parallel lines. Since each of the angles A, B, C, D, E and F are 120 degrees, the four triangles we have made are all 30-60-90 triangles. The sides of a 30-60-90 triangle can be written using  the ratio 1:√(3):2. AT = 7.5*√3 and ME = 2*√3, so the distance between the sides of length 10 and 18 is 9.5*√3. We know that DN = 3.5*√3, so BP = (9.5*√3) (3.5*√3) = 6*√3. Since the ratios of our sides should be 1:√(3):2, side BC = 2*6 = 12. Again, the answer is C. Lets look at our third and final method for solving this problem: Method 3: Using Equilateral Triangles First, extend the sides of the hexagon as shown to form a triangle: Since each internal angle of the hexagon is 120 degrees, each external angle will be 60 degrees. In that case, each angle between the dotted lines will become 60 degrees too, and hence, triangle PAB becomes  an equilateral triangle. This means PA = PB = 10. Triangle QFE   and triangle RDC also become equilateral triangles, so QF = QE = 4, and  RD = RC = 7. Now note that since angles P, Q, and R are all 60 degrees, triangle PQR is also equilateral, and hence, PQ = PR. PQ = 10 + 15 + 4 = 29 PR = 10 + BC + 7 = 29 BC = 12 (again, answer choice C) Note the geometry concepts that we used to solve this problem: regular polygon, parallel lines, angles, 30-60-90 right triangles, and equilateral triangles. We know all of these concepts very well individually, but applying them to a GMAT question can take some ingenuity! Getting ready to take the GMAT? We have  free online GMAT seminars  running all the time. And, be sure to follow us on  Facebook,  YouTube,  Google+, and  Twitter! Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the  GMAT  for Veritas Prep and regularly participates in content development projects such as  this blog!

Analysis-Synthesis Summaries - 550 Words

Analysis-Synthesis Summaries of Readings from Text (Article Critique Sample) Content: NameProfessorCourseDate2 Analysis-Synthesis Summaries of Readings from TextIntroductionWayne Enstice and Melody Peters Drawing: Space Form Expression book explores various aspects of art and drawing defined in different chapters. For instance, the focus of chapter 3 is shape, proportion and layout of drawing (Enstice and Peters 37). The authors suggest that students should consider the outline of their paintings to ensure that it depicts the correct version of the subject. A drawing should have shapes that match the principal outlines of the object in order to portray the required layout. This helps in the easy interpretation and analysis of different aspects needed in professional art works and drawings. Chapter 3 also tackles various tips on identifying shapes using pencils or viewfinder, the use of proportioning techniques and laying out of images (Enstice and Peters 38). The authors also use illustrations to explain the application of artistic concepts in drawing in relation to shape, proportion and layout. The artworks show drawings from different angles and perspectives to suit the required components.In order to express the underlining connection with the drawing forms, the author use new vocabularies. For instance, spatial configuration is the outline of a figure determined by the organization of its portions or features. This enables an artist to see objects as flat shapes in order to draw them in correct proportion (Enstice and Peters 42). By linking designated points on objects and faces spread through a space, a characteristic shape is created. The essential concept of spatial configuration is to use part of an object to produce an accurate figure. Another new vocabulary used is the Shape and Foreshortening, which is the formation of an illusion of depth. It is related to perspective since people draw figures depending on what they see rather than constructed perception. Shape and foreshortening characterizes a line, form or object a s shorter than the authentic length in order to provide a plan impression relation to the guidelines of linear perspective. For instance, in figure 3-1, the authors illustrate the use of foreshortening to explain how shapes are determined in diagrams (Enstice and Peters 44-47). The longest dimension of the body is almost perpendicular to the picture plane. Shape is achieved when the image of an object is projected onto the picture plane. This is because the shape has measurable proportions within itself that indicates different aspects. In pursuit of successfully developing skills in drawing, a student should understand how to utilize space in paintings. This essential element provides for other forms to be applied in the correct designs. The correct use of space will help one to draw dimensional shapes without problems of proportion or layout.Conclus...