{"id":12944,"date":"2021-06-13T23:06:00","date_gmt":"2021-06-13T17:36:00","guid":{"rendered":"https:\/\/e-gmat.com\/blogs\/?p=12944"},"modified":"2022-11-23T16:38:03","modified_gmt":"2022-11-23T11:08:03","slug":"geometry-practice-problems-gmat-special-properties-of-triangles","status":"publish","type":"post","link":"https:\/\/e-gmat.com\/blogs\/geometry-practice-problems-gmat-special-properties-of-triangles\/","title":{"rendered":"GMAT Geometry Practice Problems | Special Properties of Triangles (Part 3)"},"content":{"rendered":"<span class=\"rt-reading-time\" style=\"display: block;\"><span class=\"rt-label rt-prefix\">A <\/span> <span class=\"rt-time\">8<\/span> <span class=\"rt-label rt-postfix\">min read <\/span><\/span><p>In the last article in the GMAT Geometry Series, we explained properties of different types of triangles and how those properties will help you solve GMAT geometry questions on triangles. We illustrated them with GMAT geometry practice problems and also asked you to solve some GMAT Geometry Practice Problems.<\/p>\n<p><img loading=\"lazy\" class=\"wp-image-13236 alignright\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/geometry-practice-questions-gmat-geometry-problems.jpg\" alt=\"\" width=\"497\" height=\"315\" srcset=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/geometry-practice-questions-gmat-geometry-problems.jpg 1078w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/geometry-practice-questions-gmat-geometry-problems-300x190.jpg 300w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/geometry-practice-questions-gmat-geometry-problems-768x487.jpg 768w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/geometry-practice-questions-gmat-geometry-problems-1024x649.jpg 1024w\" sizes=\"(max-width: 497px) 100vw, 497px\" \/><\/p>\n<p>In this article, we will look at three special properties which are tested by GMAT in Geometry:<\/p>\n<ul>\n<li><a href=\"#property-1\">How a median affects the area of any triangle<\/a><\/li>\n<li><a href=\"#property-2\">Sum of External Angles of a triangle<\/a><\/li>\n<li><a href=\"#property-3\">Angle Side Property of Right Angled Triangle<\/a><\/li>\n<li><a href=\"#practice-problems\">GMAT Geometry Practice Problems with Detailed Solutions<\/a><\/li>\n<li><a href=\"#takeaways\">Takeaways &amp; G<span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;geometry sample problems&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:8395331,&quot;3&quot;:[null,0],&quot;4&quot;:[null,2,16777215],&quot;9&quot;:0,&quot;12&quot;:0,&quot;14&quot;:[null,2,0],&quot;15&quot;:&quot;Nunito&quot;,&quot;26&quot;:400}\">eometry Sample Problems<\/span><\/a><\/li>\n<\/ul>\n<blockquote><p><em>If you haven\u2019t gone through the previous articles of this series, here are the links-<\/em><\/p>\n<ol>\n<li><em><strong><a href=\"https:\/\/e-gmat.com\/blogs\/most-common-mistakes-in-gmat-geometry-questions-gmat-quant-prep\/\" data-wpel-link=\"internal\" target=\"_blank\" rel=\"noopener noreferrer\">Most Common Mistakes in GMAT Geometry Questions<\/a><\/strong><\/em><\/li>\n<li><em><strong><a href=\"https:\/\/e-gmat.com\/blogs\/gmat-geometry-formulas-concepts-on-triangles-part-1\/\" data-wpel-link=\"internal\" target=\"_blank\" rel=\"noopener noreferrer\">GMAT Geometry Formulas and Concepts on Triangles &#8211; Part 1<\/a><\/strong><\/em><\/li>\n<li><em><strong><a href=\"https:\/\/e-gmat.com\/blogs\/geometry-practice-questions-properties-of-triangles\/\" target=\"_blank\" rel=\"noopener noreferrer\" data-wpel-link=\"internal\">Properties of Triangles &amp; GMAT Geometry Practice Problems &#8211; Part 2<\/a><\/strong><\/em><\/li>\n<\/ol>\n<\/blockquote>\n<h2><span id=\"property-1\">Learn how a median affects the area of any triangle<\/span><\/h2>\n<p>Consider any triangle ABC and draw a line joining A to the midpoint of BC. Let\u2019s call this line AD. Now tell me, what can we infer about the area of triangle ABD and ACD?<\/p>\n<p>Is there any relation between them?<\/p>\n<p>We know,<img loading=\"lazy\" class=\"size-full wp-image-12945 alignright\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles.png\" alt=\"gmat geometry practice problems sample problems quant prep\" width=\"257\" height=\"196\"><\/p>\n<p><strong>Area of a triangle = (1\/2) x base x height<\/strong><strong>&nbsp;<\/strong><\/p>\n<p>So, let us drop a perpendicular from A on BC and name it AE.<\/p>\n<p>Since the heights of triangle ABD and ACD are equal to AE, we can write:<\/p>\n<p><strong>Area of triangle ABD = \u00bd x BD x AE<\/strong><\/p>\n<p><strong>Area of triangle ACD = \u00bd x CD x AE<\/strong><\/p>\n<p>Also, since D is the midpoint of BC, we have CD = BD<\/p>\n<p>Hence, we can conclude that:<\/p>\n<p><strong>Area of triangle ABD = Area of triangle ACD<\/strong><strong>&nbsp;<\/strong><\/p>\n<blockquote><p><strong><em>For any strategic advice for GMAT or MBA Admissions, write to us at&nbsp;<a href=\"mailto:acethegmat@e-gmat.com\">acethegmat@e-gmat.com<\/a>.&nbsp;<a href=\"https:\/\/e-gmat.com\/free-trial-registration\/?channel=blogsin_article&amp;utm_source=blogs&amp;utm_medium=in_article&amp;utm_campaign=free_trial&amp;utm_content=about_the_gmat&amp;utm_term=blogs_in_article\" data-wpel-link=\"external\" target=\"_blank\" rel=\"external noopener noreferrer\">Sign up for free trial&nbsp;<\/a><\/em><em>and get unlimited access to concept files, live sessions, and practice questions.<\/em><\/strong><\/p><\/blockquote>\n<p>We know that the line joining a vertex of a triangle to the midpoint of the opposite side is called a median of the triangle.<\/p>\n<p>Therefore, we found that the median AD divides the triangle into two equal areas.<\/p>\n<p>Let us look at an Official question which can be solved very quickly, using the above property.<\/p>\n<h5>GMAT Quant Official Guide Question | (GMAT Data Sufficiency)<\/h5>\n<p><img loading=\"lazy\" class=\"size-medium wp-image-12946 alignright\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-1-300x143.png\" alt=\"gmat geometry practice problems sample problems quant prep\" width=\"300\" height=\"143\" srcset=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-1-300x143.png 300w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-1.png 453w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><strong><u>A slightly edited version of OG Question: OG 16 DS 126<\/u><\/strong><\/p>\n<p>In triangle ABC, point X is the midpoint of side AC and point Y is the midpoint of side BC. If point R is the midpoint of line segment XC and if point S is the midpoint of line segment YC, what is the area of triangular region RCS, if the area of triangular region ABX is 32 square units?<\/p>\n<p>This diagram might look scary and the question might seem undoable at the beginning. But let us simplify the diagram, concentrating only on the medians, BX, XY, RS and RY (join RY in the diagram). In doing so, the diagram will look as follows:<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-12947 size-full\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-2.png\" alt=\"gmat geometry practice problems sample problems quant prep\" width=\"496\" height=\"150\" srcset=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-2.png 496w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-2-300x91.png 300w\" sizes=\"(max-width: 496px) 100vw, 496px\" \/><\/p>\n<p><strong><u>Solution:<\/u><\/strong><\/p>\n<p>Since BX is a median in triangle ABC, we can use the Special Property to conclude that<\/p>\n<p><strong>Area of ABX = Area of BXC<\/strong><\/p>\n<p>It is given that the Area of triangle ABX = 32 square units.<\/p>\n<p>Hence, Area of BXC = 32 square units.<\/p>\n<p>Since XY is a median in triangle BXC, we can conclude that<\/p>\n<p><strong>Area of BXY = Area of XYC = 32\/2 = 16<\/strong><\/p>\n<p>Since RY is a median in triangle XYC, we can conclude that<\/p>\n<p><strong>Area of RYX = Area of RYC = 16\/2 = 8<\/strong><\/p>\n<p>And since RS is a median in triangle YRC we can conclude that<\/p>\n<p><strong>Area of YRS = Area of RCS = 8\/2 = 4<\/strong><strong>&nbsp;<\/strong><\/p>\n<p>So, is this question really tough? Definitely, not as tough as it looked at the beginning.<\/p>\n<p>We used one simple and special property to elegantly solve a complicated looking question.<\/p>\n<h2><span id=\"property-2\">Discover the Property of Sum of External Angles of a triangle <\/span><\/h2>\n<p><img loading=\"lazy\" class=\"size-medium wp-image-12948 alignright\" style=\"color: #666666; font-size: 14px;\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-3-300x139.png\" alt=\"gmat geometry practice problems sample problems quant prep\" width=\"300\" height=\"139\" srcset=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-3-300x139.png 300w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-3.png 659w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><br \/>\nWe know that,<\/p>\n<blockquote>\n<h6><strong><em>An external angle = sum of two opposite interior angles<\/em><\/strong><\/h6>\n<\/blockquote>\n<p>In the above diagram,<\/p>\n<p>ABC is a triangle with sides extended as shown.<\/p>\n<p>Therefore, using the above property we can write,<\/p>\n<p><strong><em>Ext.&nbsp;\u2220 ACD = int.&nbsp;\u2220CAB + int. \u2220CBA<\/em><\/strong><em>&nbsp;<\/em><\/p>\n<p>But hold on, that is not the special property!<\/p>\n<p>Now you know that <em>the sum of internal angles = 180\u00b0<\/em><em>.<\/em><em>&nbsp;<\/em><\/p>\n<h6>But what about the sum of external angles?<\/h6>\n<p>If we add up all the external angles as shown in the diagram, we will get<\/p>\n<p><strong><em>Ext&nbsp;\u2220 (ACD + CAE + ABF) = int.&nbsp;\u2220 (2A + 2B + 2C) = 2 x 180\u00b0<\/em><em>&nbsp;= 360\u00b0<\/em><\/strong><em>&nbsp;<\/em><\/p>\n<p>Hence, the sum of external angles = 360<em>\u00b0<\/em><\/p>\n<p>We will discuss this property in more details for Polygons, where I will show you that sum of external angles is equal to 360<em>\u00b0<\/em> for any convex polygon, irrespective of the number of sides the polygon has.<\/p>\n<h2><span id=\"property-3\">A Cheeky Trick on Angle-Side Property of Right Angled Triangle <\/span><\/h2>\n<p>While going through GMAT geometry practice problems on right-angled triangles, the most commonly tested right-angled triangles are ones with the following set of values as its angles.<\/p>\n<ol>\n<li>30<em>\u00b0<\/em> &#8211; 60<em>\u00b0<\/em> &#8211; 90<em>\u00b0<\/em><\/li>\n<\/ol>\n<p>or<\/p>\n<ol start=\"2\">\n<li>45<em>\u00b0<\/em> &#8211; 45<em>\u00b0<\/em> &#8211; 90<em>\u00b0<\/em><\/li>\n<\/ol>\n<p>Therefore, it is important for us to know the relation between these angles and the sides of the triangles, as knowing this can help us in finding the answer quickly and at the same make the calculations significantly simple.<\/p>\n<p><img loading=\"lazy\" class=\" wp-image-12949 alignright\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-4-150x150.png\" alt=\"gmat geometry practice problems sample problems quant prep\" width=\"149\" height=\"149\" srcset=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-4-150x150.png 150w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-4-300x300.png 300w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-4.png 366w\" sizes=\"(max-width: 149px) 100vw, 149px\" \/><\/p>\n<p><strong>Case 1<\/strong>:<\/p>\n<p>As shown in the figure, if the angles of the triangle are 30<em>\u00b0<\/em>-60<em>\u00b0<\/em>-90<em>\u00b0<\/em>, the ratio of the sides will be \u2013<\/p>\n<p><strong>AB: BC: AC = 1: \u221a3: 2<\/strong><\/p>\n<p><img loading=\"lazy\" class=\"alignright wp-image-12950 \" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-5-218x300.png\" alt=\"gmat geometry practice problems sample problems quant prep\" width=\"130\" height=\"179\" srcset=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-5-218x300.png 218w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-5.png 353w\" sizes=\"(max-width: 130px) 100vw, 130px\" \/><\/p>\n<p><strong>Case 2<\/strong>:<\/p>\n<p>As shown in the figure, if the angles of the triangle are 45<em>\u00b0<\/em>-45<em>\u00b0<\/em>-90<em>\u00b0<\/em>, the ratio of the sides will be \u2013<\/p>\n<p><strong>AB: BC: AC = 1: 1: \u221a2<\/strong><\/p>\n<h2>Application of Sum of External Angles and Right Angle<\/h2>\n<p>Let us apply the special property 2 and 3 in a very simple GMAT-like question.<\/p>\n<p>The two questions given below are based on the same diagram, so let\u2019s solve these questions together to understand the application of the above properties.<\/p>\n<p><img loading=\"lazy\" class=\"alignright wp-image-12951\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-6.png\" alt=\"gmat geometry practice problems sample problems quant prep\" width=\"447\" height=\"258\" srcset=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-6.png 580w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-6-300x173.png 300w\" sizes=\"(max-width: 447px) 100vw, 447px\" \/><\/p>\n<p><strong>Q.1&nbsp;&nbsp;&nbsp; What is the value of z?<\/strong><\/p>\n<ol>\n<li>110<em>\u00b0<\/em><\/li>\n<li>120<em>\u00b0<\/em><\/li>\n<li>130<em>\u00b0<\/em><\/li>\n<li>140<em>\u00b0<\/em><\/li>\n<li>150<em>\u00b0<\/em><\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p><strong>&nbsp;Q.2&nbsp; &nbsp; If the value of AC = 4 units, what is the ratio of the sides AB and B<\/strong>C?<\/p>\n<ol>\n<li>&nbsp;2 : 1<\/li>\n<li>&nbsp;1 : \u221a 3<\/li>\n<li>&nbsp;2 : \u221a 3<\/li>\n<li>&nbsp;\u221a 3 : 1<\/li>\n<li>&nbsp;1 : 2<\/li>\n<\/ol>\n<p><strong><u>Detailed Solution to the above GMAT Geometry Practice Problem:<\/u><\/strong><strong>&nbsp;<\/strong><\/p>\n<p><strong><u>Part 1:<\/u><\/strong><\/p>\n<p>Notice carefully, that z is the external angle of triangle ABC.<\/p>\n<p>From our conceptual understanding related to angles of a triangle, we can write \u2013<\/p>\n<p><strong>External \u2220CAE = sum of opposite internal angles of triangle ABC<\/strong><\/p>\n<p>Therefore, \u2220<strong>z = \u2220ABC + \u2220ACB<\/strong><\/p>\n<p>From the diagram, we can clearly infer that \u2013<\/p>\n<p><strong>\u2220ABC = 90<em>\u00b0<\/em> and \u2220ACB = y<em>\u00b0<\/em><\/strong><\/p>\n<p>Also, \u2220ACB + \u2220ACD = 180<em>\u00b0<\/em> [straight line angle property] And, \u2220ACD = 150<em>\u00b0<\/em><\/p>\n<p>Therefore, \u2220ACB = 180<em>\u00b0<\/em> \u2013 150<em>\u00b0<\/em> = 30<em>\u00b0<\/em><\/p>\n<p>Hence, <strong>the value of \u2220z = \u2220ABC + \u2220ACB = 90<em>\u00b0<\/em><\/strong><strong>&nbsp;<\/strong><strong>+ 30<em>\u00b0<\/em><\/strong><strong>&nbsp;<\/strong><strong>= 120<em>\u00b0<\/em><\/strong><\/p>\n<p><strong>Part 2:<\/strong><\/p>\n<p>This question can be solved in multiple ways \u2013<\/p>\n<p><strong><u>Method I:<\/u><\/strong><\/p>\n<p>We know that<\/p>\n<ul>\n<li>AC = 4 units<\/li>\n<\/ul>\n<p>and<\/p>\n<ul>\n<li>from our above discussion in part 1 we know that \u2220ACB = 30<em>\u00b0<\/em><\/li>\n<\/ul>\n<p>Therefore,<\/p>\n<p>length <strong>AB = AC sin 30<em>\u00b0<\/em><\/strong><strong>&nbsp;<\/strong><strong>= 4 x \u00bd = 2<\/strong><strong>&nbsp;<\/strong><\/p>\n<p><strong>And BC = AC cos 30<em>\u00b0<\/em> = 4 x \u221a3\/2 = 2\u221a3&nbsp;<\/strong><\/p>\n<p>Therefore, ratio of <strong><u>AB : BC = 2 : 2\u221a3 = 1 : \u221a3<\/u><\/strong><\/p>\n<p><strong><u>Method II:<\/u><\/strong><strong>&nbsp;<\/strong><\/p>\n<p>But say one does not know trigonometry but still wants to solve it. This is where, our<\/p>\n<h6><strong>Special Property 3, Case 1 <\/strong>comes in handy!<\/h6>\n<ul>\n<li>If we observe carefully, we will notice that triangle ABC is a 30<em>\u00b0<\/em>-60<em>\u00b0<\/em>-90<em>\u00b0&nbsp;<\/em>triangle<\/li>\n<\/ul>\n<ul>\n<li>Since \u2220ABC = 90<em>\u00b0<\/em> , \u2220BAC = 60<em>\u00b0<\/em> and \u2220ACB = 30<em>\u00b0<\/em><\/li>\n<\/ul>\n<ul>\n<li>And we know that ratio of <strong>AB : BC : CA = 1 : \u221a3 : 2<\/strong><\/li>\n<\/ul>\n<p>Therefore, from here, we can directly say that the answer would be 1 : \u221a3 and <strong><u>we don\u2019t even have&nbsp;<\/u><\/strong><strong><u>to use the value of AC to find the answer!<\/u><\/strong><\/p>\n<p>Cool, isn\u2019t it?<\/p>\n<h2><span id=\"practice-problems\">GMAT Geometry Practice Problems &#8211; Another Question to test your imagination<\/span><\/h2>\n<p>A 60 feet long ladder is inclined against a wall such that the bottom and the top of the ladder are at equal distance from the point where the wall meets the floor. The same ladder slides away from the foot of the wall such that it is inclined at an angle of 30<em>\u00b0<\/em>&nbsp;from the floor. What is the difference between the heights to which the ladder reaches in the two cases?<\/p>\n<p><img loading=\"lazy\" class=\"alignright wp-image-12952 size-medium\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-7-300x233.png\" alt=\"GMAT Geometry Practice Problems sample problems quant prep\" width=\"300\" height=\"233\" srcset=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-7-300x233.png 300w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-7.png 614w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<ol>\n<li>15(\u221a 2 \u2013 1)<\/li>\n<li>30(\u221a 2 \u2013 1)<\/li>\n<li>15(\u221a 2 + 1)<\/li>\n<li>30\u221a 2<\/li>\n<li>30(\u221a 2 + 1)<\/li>\n<\/ol>\n<p><strong><u>Solution:<\/u><\/strong><strong>&nbsp;<\/strong><\/p>\n<p><strong><u>Approach<\/u><\/strong><strong>&nbsp;<\/strong><\/p>\n<p><em>Now this question might seem to be a tough one because the position of the ladder is different in the two cases, which leads to change in the angles with respect to the floor. Therefore, most students might feel that this question might not be an easy one to solve.<\/em><\/p>\n<p>But no need to worry, let\u2019s see how by using our special property and a methodical approach, we can get our answer very easily!<\/p>\n<h6><u>Step 1: Drawing Inferences from the Question Statement<\/u><\/h6>\n<p>Let us represent the two cases diagrammatically.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-12953\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-8.png\" alt=\"gmat geometry practice problems sample problems quant prep\" width=\"411\" height=\"196\" srcset=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-8.png 411w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-8-300x143.png 300w\" sizes=\"(max-width: 411px) 100vw, 411px\" \/><\/p>\n<p>Triangle BAC represents the initial case, where point A is the point of intersection of the wall and the floor and BC represents the ladder.<\/p>\n<p>We are given that <strong>AB = AC <\/strong>and <strong>BC = 60 feet<\/strong>.<\/p>\n<p>Triangle PAQ represents the case after the ladder slid. The only value that remains unchanged is the length of the ladder, hence BC = PQ.<\/p>\n<p>Therefore, in the Right Triangle PAQ,<\/p>\n<p>PQ = 60 feet, and \u2220AQP = 30\u00b0<strong>&nbsp;<\/strong><\/p>\n<p>We need to find the value of <strong>AB \u2013 AP.<\/strong><strong>&nbsp;<\/strong><\/p>\n<h6><u>Step 2: Finding the required values<\/u><strong>&nbsp;<\/strong><\/h6>\n<p>Let us first consider Right Triangle BAC:<\/p>\n<p>We know that when two sides of a right triangle are equal, it is a 45\u00b0-45\u00b0-90\u00b0 triangle.<\/p>\n<p>As discussed in special property 3, in such a triangle, the sides opposite to the angles 45\u00b0, 45\u00b0, and 90\u00b0 respectively are in the ratio 1: 1: \u221a2<\/p>\n<p>Therefore,<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-12954\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-9.png\" alt=\"gmat geometry practice problems sample problems quant prep\" width=\"56\" height=\"35\">&nbsp; &nbsp; &nbsp; &nbsp;\u27f9 &nbsp;&nbsp;<img loading=\"lazy\" class=\"alignnone size-full wp-image-12955\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-10.png\" alt=\"gmat geometry practice problems sample problems quant prep\" width=\"147\" height=\"36\"><\/p>\n<p>Thus, we have found out the value of AB.<\/p>\n<p>Now, considering Right Triangle PAQ:<\/p>\n<p>It is a 30\u00b0-60\u00b0-90\u00b0 right-angled triangle. And we know that in such a triangle, the sides opposite to the angles 30\u00b0, 60\u00b0, and 90\u00b0 respectively are in the ratio 1 : \u221a3 : 2.<\/p>\n<p>Therefore,<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-12956\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-11.png\" alt=\"gmat geometry practice problems sample problems quant prep\" width=\"47\" height=\"54\"><\/p>\n<p>\u27f9&nbsp;<img loading=\"lazy\" class=\"alignnone size-full wp-image-12957\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/GMAT-geometry-practice-problems-sample-problems-Quant-Prep-Triangles-12.png\" alt=\"gmat geometry practice problems sample problems quant prep\" width=\"128\" height=\"33\"><\/p>\n<h3><u>Step 3: Calculate the final answer<\/u><\/h3>\n<p>Difference between the heights to which the ladder reaches in the two cases<\/p>\n<p>= AB \u2013 AP<\/p>\n<p>= 30\u221a2 &#8211; 30<\/p>\n<p>= 30(\u221a2 \u2013 1)<\/p>\n<p>Answer: Option (2)<\/p>\n<h2><span id=\"takeaways\">Takeaways &amp; GMAT Sample Problems<\/span><\/h2>\n<p>Any geometry question testing the concepts of triangles can be easily tackled as long as the test taker is clear with:<\/p>\n<ol>\n<li><em>the basic properties of a triangle<\/em><\/li>\n<li><em>the subtle differences these properties exhibit, depending on the type of the triangle<\/em><\/li>\n<li><em>when to apply and how to apply these (Focus on the questions that are given in the article and understand how each of these properties is used in each question)<\/em><\/li>\n<\/ol>\n<blockquote>\n<h5><a href=\"https:\/\/e-gmat.com\/blogs\/gmat-online\/gmat-prep\/gmat-quant\/number-properties\/\" target=\"_blank\" rel=\"noopener noreferrer\" data-wpel-link=\"internal\"><em><strong>Access our Next Series of articles on Number Properties!<\/strong><\/em><\/a><\/h5>\n<ol>\n<li><a href=\"https:\/\/e-gmat.com\/blogs\/solve-gmat-inequalities-questions-gmat-quant\/\" target=\"_blank\" rel=\"noopener noreferrer\" data-wpel-link=\"internal\">Solving GMAT Inequalities Questions using Wavy Line Method<\/a><\/li>\n<li><a href=\"https:\/\/e-gmat.com\/blogs\/gmat-number-properties-3-mistakes-in-lcm-gcd-questions\/\" target=\"_blank\" rel=\"noopener noreferrer\" data-wpel-link=\"internal\">Avoid these 3 mistakes in LCM GCD Questions<\/a><\/li>\n<li><a href=\"https:\/\/e-gmat.com\/blogs\/gmat-quant-3-mistakes-in-even-odd-number-properties-questions\/\" target=\"_blank\" rel=\"noopener noreferrer\" data-wpel-link=\"internal\">Avoid these 3 mistakes in Even-Odd Questions<\/a><\/li>\n<\/ol>\n<\/blockquote>\n<p>With this, we come to the end of our series on GMAT Geometry. We hope you liked our articles as much as we liked creating them. Remember the takeaways from our articles on GMAT Geometry and go through GMAT geometry sample problems and geometry practice problems.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the last article in the GMAT Geometry Series, we explained properties of different types of triangles and how those properties will help you solve GMAT geometry questions on triangles. We illustrated them with GMAT geometry practice problems and also asked you to solve some GMAT Geometry Practice Problems. In this article, we will look [&hellip;]<\/p>\n","protected":false},"author":102406,"featured_media":13236,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","ub_ctt_via":""},"categories":[104,94,60],"tags":[],"featured_image_src":"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/06\/geometry-practice-questions-gmat-geometry-problems.jpg","author_info":{"display_name":"Suheb Hussain","author_link":"https:\/\/e-gmat.com\/blogs\/author\/syede-gmat-com\/"},"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.1.1 (Yoast SEO v17.1) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>GMAT Geometry Practice Problems | Special Properties of Triangles (Part 3)<\/title>\n<meta name=\"description\" content=\"GMAT Quant Prep, In this third article on GMAT geometry, Learn Special properties of triangles. 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