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href="#" class="mark-page-favorite pull-right" data-pid="2120" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1070/geometry-and-trigonometry.html">Geometry and Trigonometry</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray"> 3.5 & 3.6 Perpendicular Bisectors & Voronoi Diagrams </span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 45 minutes"><i class="fa fa-clock-o"></i> 45'</span> </ol> <article id="main-article"> <div class="intro-card"> <div class="bg-cover" style="background-image: url('/media/mathsapplications/images/vornoi.png');"> </div> <img src="../../images/vornoi-1.png" style="display: none"> <div class="content"> <h3 class="theading">Voronoi Diagrams and Perpendicular Bisectors 3.6</h3> <p class="text">In this fascinating topic, we will delve into the wonderful world of Voronoi diagrams. This includes creating Voronoi diagrams with perpendicular bisectors, adding a new site, nearest neighbour interpolation and the toxic waste dump problem. I hope you enjoy applying mathematics to the real world in this rich and interesting area of the course</p> </div> </div> <div class="panel panel-has-colored-body panel-turquoise"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key concepts</p> </div> </div> <div class="panel-body"> <p><span id="docs-internal-guid-8648c51e-7fff-465f-b18d-1d820025c0b9"></span></p> <p dir="ltr"><span id="docs-internal-guid-8648c51e-7fff-465f-b18d-1d820025c0b9">In this unit you should learn to…</span></p> <ul style="list-style-type:disc;"> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Be able to find perpendicular gradients and hence a perpendicular bisector between two points. </p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Be able to create a Voronoi diagram with two or three sites. </p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Be able to add a new site to a Voronoi diagram. </p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Be able use nearest neighbour interpolation. </p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Be able to find the largest empty circle in a Voronoi diagram. </p> </li> </ul> </div> </div> <div class="panel panel-has-colored-body panel-orange"> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. Introducing Voronoi Diagrams</p> </div> </div> <div class="panel-body"> <div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/478859573"></iframe></div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. Perpendicular Gradients</p> </div> </div> <div class="panel-body"> <div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/478861181"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Perpendicular Bisectors</p> </div> </div> <div class="panel-body"> <div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/478863303"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p> 4. Creating a Voronoi Diagram with Two Sites</p> </div> </div> <div class="panel-body"> <div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/480027494"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>5. Creating a Voronoi Diagram with Three Sites</p> </div> </div> <div class="panel-body"> <div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/480028758"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p> 6. Adding a New Site to a Voronoi Diagram</p> </div> </div> <div class="panel-body"> <div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/480029739"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>7. The Nearest Neighbour Algorithm</p> </div> </div> <div class="panel-body"> <div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/482304404"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p> 8. The Toxic Waste Dump Problem</p> </div> </div> <div class="panel-body"> <div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/482305196"></iframe></div> </div> </div> </div> </div> </div> <div class="panel panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div id="carousel-271" class="dynamic-gallery carousel slide" data-id="271"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-271/flashcard1.png" data-fancybox="gallery-271" title="" data-caption=""><img alt="" src="../../../std-galleries/15-271/flashcard1.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-271/flashcard2.png" data-fancybox="gallery-271" title="" data-caption=""><img alt="" src="../../../std-galleries/15-271/flashcard2.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-271/flashcard3.png" data-fancybox="gallery-271" title="" data-caption=""><img alt="" src="../../../std-galleries/15-271/flashcard3.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-271/flashcard4.png" data-fancybox="gallery-271" title="" data-caption=""><img alt="" src="../../../std-galleries/15-271/flashcard4.png"></a></div></div><a class="left carousel-control" href="#carousel-271" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-271" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-271/flashcard1-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-271/flashcard2-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-271/flashcard3-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/15-271/flashcard4-thumb128.jpg"><li></li></ol> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test yourself</p> </div> </div> <div class="panel-body"> <div> <h3>Quiz</h3> <p>Practice your understanding on these quiz questions. Check your answers when you are done and read the hints where you got stuck. If you find there are still some gaps in your understanding then go back to the videos and slides above.</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="tib-quiz" data-stats="15-574-2120"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the gradient of the perpendicular bisector of the points (1,1) and (7,3)?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{3}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-3\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\frac{1}{2}\)</span></label></p></div><div class="q-explanation"><p>The gradient between the two points is <span class="math-tex">\(\frac{1}{3}\)</span>. Hence, the perpendiular gradient is <span class="math-tex">\(-\frac{3}{1}=-3\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the equation of the perpendicular bisector between the points (-2, 5) and (4, 3).</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=\frac{1}{3}x-\frac{1}{3}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=\frac{1}{3}x+3\frac{2}{3}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(y=3x+1\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=3x-11\)</span></span></label> </p></div><div class="q-explanation"><p>Midpoint: <span class="math-tex">\(\left(\frac{-2+4}{2},\frac{5+3}{2}\right)=\left(1,4\right)\)</span></p><p>Gradient: <span class="math-tex">\(\frac{3-5}{4--2}=-\frac{2}{6}=-\frac{1}{3}\)</span></p><p>Perpendicular Gradient: <span class="math-tex">\(\frac{3}{1}=3\)</span></p><p>Equation: <span class="math-tex">\(y=3x+c\)</span></p><p>Substitute <span class="math-tex">\((1,4)\)</span> into the equation: <span class="math-tex">\(4=3\times1+c\)</span>, so <span class="math-tex">\(c=1\)</span>.</p><p>Thus, <span class="math-tex">\(y=3x+1\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the equation of the perpendicular bisector between the points given on the graph below. </p><p><img alt="" src="../../images/picture1(1).png" style="width: 250px; height: 256px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=0x+4\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=4\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(x =4\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=x+4\)</span></span></label> </p></div><div class="q-explanation"><p>A vertical line which passes through <span class="math-tex">\(4\)</span> on the x-axis has equation, <span class="math-tex">\(x =4\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>A line segment has a gradient of <span class="math-tex">\(3\)</span>. What is the gradient of the perpendiular line?</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(-\frac{1}{3}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-0.3\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{1}{3}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-3\)</span></span></label> </p></div><div class="q-explanation"><p>If the gradient is <span class="math-tex">\(3\)</span>, we can write the gradient as <span class="math-tex">\(\frac{3}{1}\)</span> so that the perpendicular gradient is <span class="math-tex">\(-\frac{1}{3}\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Below is a Voronoi diagram with an edge missing. Find the equation of the edge, and write your answer in the form <span class="math-tex">\(ax+by+c=0\)</span>, where <span class="math-tex">\(a,b,c \in \mathbb{Z}\)</span> and <span class="math-tex">\(a,b>0\)</span>.</p><p><img alt="" src="../../images/voronoipi.png" style="width: 250px; height: 260px;"></p></div><div class="q-answer"><p>Equation: <input type="text" style="height: auto;" data-c="10x+4y-55=0"> <span class="review"></span> </p></div><div class="q-explanation"><p>The missing edge seperates sites A(1,4) and B(6,6). </p><p>Midpoint: <span class="math-tex">\(\left(\frac{1+6}{2},\frac{4+6}{2}\right)=\left(3.5,5\right)\)</span></p><p>Gradient: <span class="math-tex">\(\frac{6-4}{6-1}=\frac{2}{5}\)</span></p><p>Perpendicular Gradient: <span class="math-tex">\(-\frac{5}{2}\)</span></p><p>Equation: <span class="math-tex">\(y=-\frac{5}{2}x+c\)</span></p><p>Substitute <span class="math-tex">\((3.5,5)\)</span> into the equation: <span class="math-tex">\(5=3.5\times-\frac{5}{2}+c\)</span>, so <span class="math-tex">\(c=13.75\)</span>.</p><p>Thus, <span class="math-tex">\(y=-\frac{5}{2}x+13.75\)</span></p><p>Since <span class="math-tex">\(a,b,c \)</span> must be integers with <span class="math-tex">\(a,b>0\)</span>, we can multiply te equation by <span class="math-tex">\(4\)</span> and rearrange to obtain </p><p><span class="math-tex">\(10x+4y-55=0\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>The rainfall in three nearby cities, A, B, C and D, over the course of the previous month, is 15 cm, 22 cm, 18 cm and 20 cm respectively. Use nearest neighbour interpolation to estimate the rainfall at (2, 3).</p><p><img alt="" src="../../images/picture1(2).png" style="width: 300px; height: 242px;"></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="19"> <span class="review"></span> cm</p></div><div class="q-explanation"><p>The point (2, 3) is located on an edge between the sites C and D. We therefore take the mean of the rainfalls at C and D: <span class="math-tex">\(\frac{18+20}{2}=19\ cm\)</span></p><p>If the point was located inside a cell, then we would estimate the rainfall to be the same as the site within that cell. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>The rainfall in three nearby cities, A, B, C and D, over the course of the previous month, is 15 cm, 22 cm, 18 cm and 20 cm respectively. Use nearest neighbour interpolation to estimate the rainfall at (5, 3).</p><p><img alt="" src="../../images/picture1(3).png" style="width: 300px; height: 242px;"></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="20"> <span class="review"></span> cm</p></div><div class="q-explanation"><p>The point (5, 3) is located on a vertex between the sites B, C and D. We therefore take the mean of the rainfalls at B, C and D: <span class="math-tex">\(\frac{22+18+20}{3}=20\ cm\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>In the Voronoi diagram below, a new site D has been added. </p><p><img alt="" src="../../images/newsite.png" style="width: 300px; height: 205px;"> </p><p>Choose which statement below is the correct process to create a new Voronoi diagram for sites A, B, C and D.</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> Find a perpendicular bisector between D and A and ensure it hits te vertex already in the current Voronoi diagram. </label></p><p><label class="checkbox"><input type="checkbox"> Find a perpendicular bisector between D and C and ensure it stops at the edge between A and C.</label></p><p><label class="checkbox"><input class="c" type="checkbox"> Find a perpendicular bisector between D and C, ensuring it stops at the edge between A and C, then find a second prpendicular bisector between D and A and construct the new Voronoi diagram accordingly. </label></p></div><div class="q-explanation"><p>When adding a new site, a perpendicular bistector needs to be found between te new site, and the site which already exists in the cell. Then a new perpendicular bistector must be found between the new site, and any sites in cells which are directly next to the new site. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the co-ordinates of the missing site in the Voronoi diagram below.</p><p><img alt="" src="../../images/picture1(4).png" style="width: 300px; height: 245px;"></p></div><div class="q-answer"><p>( <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span> , <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span> )</p></div><div class="q-explanation"><p>The missing site is contained in the cell which currently does not have a site. </p><p>If we start by looking at the edge between the missing site and site D, it has gradient of 3 right and 1 down, <span class="math-tex">\(-\frac{1}{3}\)</span>. Therefore the perpendicular gradient must be <span class="math-tex">\(3\)</span>. Utilising the same idea with the edge between site B and the missing site, the gradient is <span class="math-tex">\(-\frac{1}{2}\)</span>, which means the perpendicular gradient must be <span class="math-tex">\(2\)</span>. Starting at site D and B, and drawing a line backwards with the perpendicular gradients found will ascertain the location of the missing site. </p><p><img alt="" src="../../images/picture1(5).png" style="width: 300px; height: 242px;"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the centre and radius of the largest empty circle in the Voronoi diagram below. </p><p><img alt="" src="../../images/picture1(6).png" style="width: 300px; height: 232px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\((5, 6); 4\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\((5, 6); \sqrt{34}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\((9, 5); \sqrt{15}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\((9, 5); \sqrt{17}\)</span></span></label> </p></div><div class="q-explanation"><p>Since there are two vertices, we will need to check the distance between each vertex and a randomly picked site that each vertex seperates. </p><p>For the vertex <span class="math-tex">\((9,5)\)</span>, if I randomly pick site A, the distance is given by, <span class="math-tex">\(d=\sqrt{(9-13)^2+(5-4)^2}=\sqrt{(-4)^2+1^2}=\sqrt{17} \)</span></p><p>For the vertex (5, 6), if I randomly pick site D, the distance is given by,</p><p><span class="math-tex">\(d=\sqrt{(5-0)^2+(6-9)^2}=\sqrt{5^2+3^2}=\sqrt{34} \)</span></p><p>Therefore, the largest empty circle has centre <span class="math-tex">\((5, 6)\)</span> and radius <span class="math-tex">\(\sqrt{34} \)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam Style Questions</p> </div> </div> <div class="panel-body"> <div> <p>The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.</p> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <p><img alt="" src="../../images/exam1_image.png" style="width: 600px; height: 513px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/500578923"></iframe></div> </section> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p> Question 2</p> </div> </div> <div class="panel-body"> <p><img alt="" src="../../images/exam2_image(1).png" style="width: 600px; height: 377px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/500579451"></iframe></div> </section> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3 </p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../images/exam3a_image.png" style="width: 600px; height: 151px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/500579893"></iframe></div> </section> </div> <p><img alt="" src="../../images/picture1(7).png" style="width: 600px; height: 137px;"></p> <p>Video solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/500580294"></iframe></div> </section> </div> <p> </p> <div class="panel panel-default panel-expandable panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <p dir="ltr"><span id="docs-internal-guid-336c7471-7fff-3ffc-d43d-968a9548160c">Gil is designing his garden ABCD, and is planning on a set of coordinate axes. Each unit represents 1 metre of his garden. </span></p> <p dir="ltr"><img alt="" src="../../screenshot-2022-04-11-at-20.46.51.png" style="width:400px;height:353px;"></p> <p dir="ltr" role="presentation"><span id="docs-internal-guid-336c7471-7fff-3ffc-d43d-968a9548160c">a. Find the area of his garden </span></p> <p dir="ltr"><span id="docs-internal-guid-336c7471-7fff-3ffc-d43d-968a9548160c">b.He is building a path (represented by EBGF). What is the gradient of the line through EB?</span></p> <p dir="ltr"><span id="docs-internal-guid-336c7471-7fff-3ffc-d43d-968a9548160c">c. What is the equation of the line through EB? Give your answer in the form ax+by+d=0</span></p> <p dir="ltr"><span id="docs-internal-guid-336c7471-7fff-3ffc-d43d-968a9548160c">d. He wants to make a new path from A that meets the original path at a right angle. 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