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Midpoints and Segment Bisectors ( Read ) | Geometry
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                <div class="title break-word titleonly">
                  Midpoints and Segment Bisectors
                </div>
               </div>
              </div>
              <div class="modalitycontent clearfix">
               <article>
                <div class="annotator-wrapper">
                 <div id="modality_content" class="modality_content modality_type_read" data-loadurl="/flx/get/perma/modality/lesson/Midpoints-and-Segment-Bisectors-BSC-GEOM/MAT.GEO.103.01?extension=version%3A23&amp;format=html">
                  <h3 id="x-ck12-R3VpZGFuY2U."><span>Midpoints and Segment Bisectors</span></h3>
                  <p id="x-ck12-YjQ4ZTQ3YmFjZmRjMGNmZjU0NWI4YzQ1MDVhYjA2ODk.-b7p">When two segments are congruent, we indicate that they are congruent, or of equal length, with <span class="x-ck12-underline"><span class="vocab_term">segment markings</span></span>, as shown below:</p>
                  <p id="x-ck12-OThlZDYwZGIxYzVmMTMzNTlmYjYxMDYwYTljOThlNDk.-sde"><span class="x-ck12-img-inline">
                    <!-- @@author="" -->
                    <!-- @@license="" --><img data-flx-url="/flx/show/image/201502261424993995157108_588ff8036484739364493a03e596e988-201502261424996108674496.png" src="https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A6a98a3ad7d0f7e30f7ac6626d413a3e250ddfed53d0b2d22ea146f08%2BIMAGE_TINY%2BIMAGE_TINY.1" width="320" alt=""></span></p>
                  <p id="x-ck12-NWZlYmFkOTUyNWQ4ODBmYjU0YmQ0OWJiYjRmZmZjZTA.-czn">A <strong><span class="vocab_term">midpoint</span></strong> is a point on a line segment that divides it into two congruent segments.</p>
                  <p id="x-ck12-OThlZDYwZGIxYzVmMTMzNTlmYjYxMDYwYTljOThlNDk.-hcq"><span class="x-ck12-img-inline">
                    <!-- @@author="" -->
                    <!-- @@license="" --><img data-flx-url="/flx/show/image/201502261424993995174918_f7544e7ededc28002f16e3f336e86e66-201502261424996108893779.png" src="https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A31d2d6b90116b83c83ccbcb107cb09938f3832d09105ace9df6427e7%2BIMAGE_TINY%2BIMAGE_TINY.1" width="150" alt=""></span></p>
                  <p id="x-ck12-MzlmMzU2MTcxZjVjM2Y1YTEwZDE4NTA0N2EzMDEzYTc.-nsk">Because <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="AB%20%3D%20BC%2C%20%5C%20B" data-math-class="x-ck12-math"></span> is the midpoint of <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%5Coverline%7BAC%7D" data-math-class="x-ck12-math"></span>. Any line segment will have exactly one midpoint.</p>
                  <p id="x-ck12-MmVjYjAyYmYyNmM0NzBkYzI1NmYwOTAyYzU0YWEzMTI.-nmc">When points are plotted in the coordinate plane, we can use a formula to find the midpoint between them.</p>
                  <p id="x-ck12-NTI3ZjNiZjRmY2E4M2U3MDVhMGY5OTUwZGEzNjFlOWE.-y6h">Here are two points, (-5, 6) and (3, 2).</p>
                  <p id="x-ck12-OThlZDYwZGIxYzVmMTMzNTlmYjYxMDYwYTljOThlNDk.-egd"><span class="x-ck12-img-inline">
                    <!-- @@author="" -->
                    <!-- @@license="" --><img data-flx-url="/flx/show/image/201502261424993995250018_d35bc7189e3b7b2c80be76902e7d6bd3-201502261424996109071515.png" src="https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A650ea721ae597cc15733b51060539384713704655785db3409ca65de%2BIMAGE_TINY%2BIMAGE_TINY.1" width="250" alt=""></span></p>
                  <p id="x-ck12-MzliMjAzNTYwYjNjN2E2YzkwMTI5MTFhYmMxYWY5N2U.-hr2">The midpoint should be halfway between the points on the segment connecting them. Just by looking, it seems like the midpoint is (-1, 4).</p>
                  <p id="x-ck12-ODU5YWNhZTYxMmNmY2I3MTVmYmU0NTY3NDdlODZhNWI.-5xd"><strong><span class="vocab_term">Midpoint Formula</span>:</strong> For two points, <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%28x_1%2C%20y_1%29" data-math-class="x-ck12-math"></span> and <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="x_2%2C%20y_2" data-math-class="x-ck12-math"></span>, the midpoint is <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%5Cleft%20%28%20%5Cfrac%7Bx_1%2Bx_2%7D%7B2%7D%2C%20%5C%20%5Cfrac%7By_1%2By_2%7D%7B2%7D%20%5Cright%20%29" data-math-class="x-ck12-math"></span>.</p>
                  <p id="x-ck12-Mzk2ZGE3NjNlOGYzODFlZDM0Y2JkMWQ0M2JhYzkzM2I.-ars">Let’s use the formula to make sure (-1, 4) is the midpoint between (-5, 6) and (3, 2).</p>
                  <p id="x-ck12-6zw"><span class="x-ck12-mathEditor" data-mathmethod="block" data-contenteditable="false" data-edithtml="" data-tex="%5Cleft%20%28%20%5Cfrac%7B-5%2B3%7D%7B2%7D%2C%20%5C%20%5Cfrac%7B6%2B2%7D%7B2%7D%20%5Cright%20%29%20%3D%20%5Cleft%20%28%5Cfrac%7B-2%7D%7B2%7D%2C%20%5Cfrac%7B8%7D%7B2%7D%20%5Cright%20%29%20%3D%20%28-1%2C%204%29" data-math-class="x-ck12-block-math"></span></p>
                  <p id="x-ck12-MTllNGVlZGVkMzFiZThmMDQ4ZGFmYWM2NGM1MTJiNDQ.-srn">A <strong><span class="vocab_term">segment bisector</span></strong> cuts a line segment into two congruent parts and passes through the midpoint. A <strong><span class="vocab_term">perpendicular bisector</span></strong> is a segment bisector that intersects the segment at a right angle.</p>
                  <p id="x-ck12-lli"><span class="x-ck12-mathEditor" data-mathmethod="block" data-contenteditable="false" data-edithtml="" data-tex="%26%5Coverline%7BAB%7D%20%5Ccong%20%5Coverline%7BBC%7D%5C%5C%0A%26%5Coverline%7BAC%7D%20%5Cperp%20%5Coverleftrightarrow%7BDE%7D" data-math-class="x-ck12-block-math"></span></p>
                  <p id="x-ck12-OThlZDYwZGIxYzVmMTMzNTlmYjYxMDYwYTljOThlNDk.-xft"><span class="x-ck12-img-inline">
                    <!-- @@author="" -->
                    <!-- @@license="" --><img data-flx-url="/flx/show/image/201502261424993995285983_00838ea2dc065d5cdcb3adf11ed6fd62-201502261424996109251396.png" src="https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A10b5c81e8971228e3765f5bf8c316678e86850a859736f82bdcaad1a%2BIMAGE_TINY%2BIMAGE_TINY.1" width="150" alt=""></span></p>
                  <p id="x-ck12-ZmYzZjdhYTEwNzdlODYwZDAwMDAyYjNiY2Q4OTVlODA.-0pg"><span>What if you were given the coordinates of two points and you wanted to find the point exactly in the middle of them? How would you find the coordinates of this third point?</span></p>
                  <p id="x-ck12-MmIyZDljNjI4YmRlZWQwYTZhZjc1YTMzYjBlZmQ4YzA.-2l7" style="display: none;">&nbsp;</p>
                  <div id="x-ck12-rzf" itemprop="video" itemscope itemtype="http://schema.org/VideoObject">
                   <div class="">
                    <iframe id="x-ck12-x-ck12-LURheWRVejMxbHc." src="/flx/show/video/http%3A//www.youtube.com/embed/-DaydUz31lw%3Fwmode%3Dtransparent%26rel%3D0%26hash%3Dd87de95619743f92c88d01b2535caa42" name="136725" frameborder="0" height="380" width="500" class="ck12-embed-iframe"></iframe>
                   </div>
                  </div>
                  <div id="x-ck12-awo" itemprop="video" itemscope itemtype="http://schema.org/VideoObject">
                   <div class="">
                    <iframe id="x-ck12-x-ck12-ZDBNaFM4RUl6RlE." src="/flx/show/cover%20video/http%3A//www.youtube.com/embed/d0MhS8EIzFQ" name="1283" frameborder="0" height="380" width="500" class="ck12-embed-iframe"></iframe>
                   </div>
                  </div>
                  <p id="x-ck12-MmIyZDljNjI4YmRlZWQwYTZhZjc1YTMzYjBlZmQ4YzA.-toy" style="display: none;">&nbsp;</p>
                  <h3>Examples</h3>
                  <h4 id="x-ck12-RXhhbXBsZSBB">Example 1</h4>
                  <p id="x-ck12-NTI3NDhmMDM3YWRmMmJjNmM1YTVjM2Q4NzAwMDIwNWE.-3h2">Write all equal segment statements.</p>
                  <p id="x-ck12-OThlZDYwZGIxYzVmMTMzNTlmYjYxMDYwYTljOThlNDk.-5qr"><span class="x-ck12-img-inline">
                    <!-- @@author="" -->
                    <!-- @@license="" --><img data-flx-url="/flx/show/image/201502261424993995318372_b6dba2a551700313344ebcb95d35a205-201502261424996109464838.png" src="https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A9a6700e4e1ad8f7c4722a6bf4668a0b40cb5ffb6bd46c24bc8a1e71d%2BIMAGE_TINY%2BIMAGE_TINY.1" width="220" alt=""></span></p>
                  <p id="x-ck12-qt6"><span class="x-ck12-mathEditor" data-mathmethod="block" data-contenteditable="false" data-edithtml="" data-tex="AD%20%26%3D%20DE%5C%5C%0AFD%20%26%3D%20DB%20%3D%20DC" data-math-class="x-ck12-block-math"></span></p>
                  <h4 id="x-ck12-RXhhbXBsZSBC">Example 2</h4>
                  <p id="x-ck12-OWI1NDRmOTU3YWY4ZGNkNGZhZjgyZTQ5ZTQ3ODUzZGQ.-kom">Is <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="M" data-math-class="x-ck12-math"></span> a midpoint of <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%5Coverline%7BAB%7D" data-math-class="x-ck12-math"></span>?</p>
                  <p id="x-ck12-OThlZDYwZGIxYzVmMTMzNTlmYjYxMDYwYTljOThlNDk.-oqw"><span class="x-ck12-img-inline">
                    <!-- @@author="" -->
                    <!-- @@license="" --><img data-flx-url="/flx/show/image/201502261424993995333147_fac7855bf22a4f859f4fa80e5d331af8-201502261424996109649685.png" src="https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A76ab1ddc9aae00b907f6e17901a9a17bb9fbf549ae9e399097fdb6fd%2BIMAGE_TINY%2BIMAGE_TINY.1" width="170" alt=""></span></p>
                  <p id="x-ck12-MzVhYTdlOTk5N2RhNjhhMDA0OWFkYTg3NzE1NWIxOWI.-btd">No, it is not <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="MB%20%3D%2016" data-math-class="x-ck12-math"></span> and <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="AM%20%3D%2034%20-%2016%20%3D%2018" data-math-class="x-ck12-math"></span>. <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="AM" data-math-class="x-ck12-math"></span> must equal <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="MB" data-math-class="x-ck12-math"></span> in order for <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="M" data-math-class="x-ck12-math"></span> to be the midpoint of <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%5Coverline%7BAB%7D" data-math-class="x-ck12-math"></span>.</p>
                  <h4 id="x-ck12-RXhhbXBsZSBD">Example 3</h4>
                  <p id="x-ck12-MDE2OTRmNTgxOGYyYjdlMDU2NmFkMzQzMDAyODI2NTM.-kxr">Find the midpoint between (9, -2) and (-5, 14).</p>
                  <p id="x-ck12-ZDY3MDRjYmRhZTgxNGNlOTQ2MDg2MjQzMWY4ZGFiMDY.-0xu">Plug the points into the formula.</p>
                  <p id="x-ck12-jde"><span class="x-ck12-mathEditor" data-mathmethod="block" data-contenteditable="false" data-edithtml="" data-tex="%5Cleft%20%28%20%5Cfrac%7B9%2B%28-5%29%7D%7B2%7D%2C%20%5Cfrac%7B-2%2B14%7D%7B2%7D%20%5Cright%20%29%20%3D%20%5Cleft%20%28%20%5Cfrac%7B4%7D%7B2%7D%2C%20%5Cfrac%7B12%7D%7B2%7D%20%5Cright%20%29%20%3D%20%282%2C%206%29" data-math-class="x-ck12-block-math"></span></p>
                  <h4 id="x-ck12-Y2JmYTg3Yzc1MzM2MjFiMzcwMWQzMjYyOWUwNDlmMDM.-nef">Example 4</h4>
                  <p id="x-ck12-NzE3ZDljNTRjZTVmYjRhMTBmMGM2NmNhZjUwMDc3Mjc.-y5w">Which line is the perpendicular bisector of <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%5Coverline%7BMN%7D" data-math-class="x-ck12-math"></span>?</p>
                  <p id="x-ck12-OThlZDYwZGIxYzVmMTMzNTlmYjYxMDYwYTljOThlNDk.-kcu"><span class="x-ck12-img-inline">
                    <!-- @@author="" -->
                    <!-- @@license="" --><img data-flx-url="/flx/show/image/201502261424993995366844_93b370aa6fe0ad4907b2c19264660b46-201502261424996109869914.png" src="https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3Aeb304884693938a38d56bff2bdfecc528f4a3e963c0091768fb3056d%2BIMAGE_TINY%2BIMAGE_TINY.1" width="150" alt=""></span></p>
                  <p id="x-ck12-NjJmMWU4ZWE2MWYwNTQyODM4NDI0MjM5MzljNWE3OTU.-fdp"><span>The perpendicular bisector must bisect</span> <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%5Coverline%7BMN%7D" data-math-class="x-ck12-math"></span> <span>and be perpendicular to it. Only</span> <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%5Coverleftrightarrow%7BOQ%7D" data-math-class="x-ck12-math"></span> <span>fits this description.</span> <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%5Coverleftrightarrow%7BSR%7D" data-math-class="x-ck12-math"></span> <span>is a bisector, but is not perpendicular.</span></p>
                  <h4 id="x-ck12-NWZhNDZjNjBiZTc2YWRlYzNhMzE3OGZhYzc2YTM0NmY.-wlv">Example 5</h4>
                  <p id="x-ck12-YTdhOGQ5YTNjYmZmMGNiYTUwODY0YjZlOTM4N2FhZDQ.-rf0">Find <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="x" data-math-class="x-ck12-math"></span> and <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="y" data-math-class="x-ck12-math"></span>.</p>
                  <p id="x-ck12-OThlZDYwZGIxYzVmMTMzNTlmYjYxMDYwYTljOThlNDk.-b5o"><span class="x-ck12-img-inline">
                    <!-- @@author="" -->
                    <!-- @@license="" --><img data-flx-url="/flx/show/image/201502261424993995394699_7a4e4c52a00491956d3320551e174af6-201502261424996110055260.png" src="https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A749cf80dba8fb29b27b801d636209e36994f8e53fbe86b145c35ff53%2BIMAGE_TINY%2BIMAGE_TINY.1" width="200" alt=""></span></p>
                  <p id="x-ck12-N2Q1YTY5Njk4MDJiYjVlMWQ5MzFiNTEwYThmZGIzYmE.-bsm">The line shown is the perpendicular bisector.</p>
                  <p id="x-ck12-6mo"><span class="x-ck12-mathEditor" data-mathmethod="block" data-contenteditable="false" data-edithtml="" data-tex="%5Ctext%7BSo%7D%2C%20%5C%203x%20-%206%20%26%3D%2021%20%26%26%20%5Ctext%7BAnd%7D%2C%20%5C%20%284y%20-%202%29%5E%5Ccirc%20%3D%2090%5E%5Ccirc%5C%5C%0A3x%20%26%3D%2027%20%26%26%20%5Cqquad%20%5Cqquad%20%5C%20%5C%20%5Cquad%204y%20%3D%2092%5E%5Ccirc%5C%5C%0Ax%20%26%3D%209%20%26%26%20%5Cqquad%20%5Cqquad%20%5Cquad%20%5C%20%5C%20%5C%20y%20%3D%2023%5E%5Ccirc" data-math-class="x-ck12-block-math"></span></p>
                  <h3 id="x-ck12-RXhwbG9yZSBNb3Jl">Review</h3>
                  <ol id="x-ck12-NTA0ZWE4NWFhMWRjOTI4NWYwNDMzNzQ1YzQ2NzMyNGM.-zy9">
                   <li>Copy the figure below and label it with the following information:</li>
                  </ol>
                  <p id="x-ck12-zni"><span class="x-ck12-mathEditor" data-mathmethod="block" data-contenteditable="false" data-edithtml="" data-tex="%5Coverline%7BAB%7D%20%26%20%5Ccong%20%5Coverline%7BCD%7D%5C%5C%0A%5Coverline%7BAD%7D%20%26%20%5Ccong%20%5Coverline%7BBC%7D" data-math-class="x-ck12-block-math"></span></p>
                  <p id="x-ck12-OThlZDYwZGIxYzVmMTMzNTlmYjYxMDYwYTljOThlNDk.-cbp"><span class="x-ck12-img-inline">
                    <!-- @@author="" -->
                    <!-- @@license="" --><img data-flx-url="/flx/show/image/201502261424993995435828_1a2dde2d8d80596b078cd8d8f07a6d44-201502261424996110275877.png" src="https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A9a867539e404bba62da8afe4e566f7b4f90a209239f13b447100eeda%2BIMAGE_TINY%2BIMAGE_TINY.1" width="100" alt=""></span></p>
                  <p id="x-ck12-OTRjMGEyZjc4ZDdiNWMwNGY3MmRjZTZkOGNiMmYyNjU.-vex">For 2-4, use the following picture to answer the questions.</p>
                  <p id="x-ck12-OThlZDYwZGIxYzVmMTMzNTlmYjYxMDYwYTljOThlNDk.-ntt"><span class="x-ck12-img-inline">
                    <!-- @@author="" -->
                    <!-- @@license="" --><img data-flx-url="/flx/show/image/201502261424993995464664_7670cdaa99c4b3e1f2cb24d319d4367f-201502261424996110464085.png" src="https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A0e813b80008999b9baa27e076d8989675b02577c40d06ffad1b1ba1c%2BIMAGE_TINY%2BIMAGE_TINY.1" width="200" alt=""></span></p>
                  <ol id="x-ck12-NjUwMmZhM2NkZGYxMjI4M2YyZTk2ZTU4MzY3MDc5MWY.-hdm" start="2">
                   <li><span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="P" data-math-class="x-ck12-math"></span> is the midpoint of what two segments?</li>
                   <li>How does <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%5Coverline%7BVS%7D" data-math-class="x-ck12-math"></span> relate to <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%5Coverline%7BQT%7D" data-math-class="x-ck12-math"></span>?</li>
                   <li>How does <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%5Coverline%7BQT%7D" data-math-class="x-ck12-math"></span> relate to <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%5Coverline%7BVS%7D" data-math-class="x-ck12-math"></span>?</li>
                  </ol>
                  <p id="x-ck12-MTkzMzA5ZmYxNDZmNmQwMGFmODFlMGY4ZTM5OGFiZmY.-u0w">For exercise 5, use algebra to determine the value of&nbsp;<span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="x." data-math-class="x-ck12-math"></span>&nbsp;</p>
                  <ol id="x-ck12-OThlZDYwZGIxYzVmMTMzNTlmYjYxMDYwYTljOThlNDk.-ezm" start="5">
                   <li><span class="x-ck12-img-inline">
                     <!-- @@author="" -->
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                  <p id="x-ck12-MTFjNDdkOTA3MDBmMTYwNDYzYmEyZWNjMTE1MzcyNDk.-rhf">For questions 6-10, find the midpoint between each pair of points.</p>
                  <ol id="x-ck12-ZjMwZTQ0NDE3MDk0NDRlMWQwMTA2YTljM2YzY2Y4M2U.-u9f" start="6">
                   <li>(-2, -3) and (8, -7)</li>
                   <li>(9, -1) and (-6, -11)</li>
                   <li>(-4, 10) and (14, 0)</li>
                   <li>(0, -5) and (-9, 9)</li>
                   <li>(-3, -5) and (2, 1)</li>
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                  <p id="x-ck12-OTQ2ZWUyYTJhZDQwOGFkYjA0MTU2N2EyNjYyODY5YmI.-vji">Given the midpoint <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%28M%29" data-math-class="x-ck12-math"></span> and either endpoint of <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="%5Coverline%7BAB%7D" data-math-class="x-ck12-math"></span>, find the other endpoint.</p>
                  <ol id="x-ck12-YjU2YTBlOGRhYmUxZTllNmE3ZmQ2OTEzZTY0NDgyMjA.-2xw" start="11">
                   <li><span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="A%28-1%2C%202%29" data-math-class="x-ck12-math"></span> and <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="M%283%2C%206%29" data-math-class="x-ck12-math"></span></li>
                   <li><span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="B%28-10%2C%20-7%29" data-math-class="x-ck12-math"></span> and <span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="M%28-2%2C%201%29" data-math-class="x-ck12-math"></span></li>
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                       <td><span class="vterm vtitle"><strong>midpoint</strong></span></td>
                       <td><span class="vdefinition">The midpoint of a line segment is the point on the line segment that splits the segment into two congruent parts.</span></td>
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                       <td><span class="vterm vtitle"><strong>perpendicular bisector</strong></span></td>
                       <td><span class="vdefinition">A segment bisector that intersects the segment at a right angle.</span></td>
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                       <td><span class="vterm vtitle"><strong>segment bisector</strong></span></td>
                       <td><span class="vdefinition">A segment bisector is a line (or part of a line) that passes through the midpoint.</span></td>
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                       <td><span class="vterm vtitle"><strong>segment markings</strong></span></td>
                       <td><span class="vdefinition">When two segments are congruent, we indicate that they are congruent with <b><i>segment markings</i></b>.</span></td>
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                       <td><span class="vterm vtitle"><strong>Midpoint Formula</strong></span></td>
                       <td><span class="vdefinition">The midpoint formula says that for endpoints <span><span class="MathJax_Preview">(x_1, y_1)</span><script type="math/tex">(x_1, y_1)</script></span> and <span><span class="MathJax_Preview">(x_2, y_2)</span><script type="math/tex">(x_2, y_2)</script></span>, the midpoint is <span><span class="MathJax_Preview"></span><script type="math/tex"></script>\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)<span class="MathJax_Preview"></span><script type="math/tex"></script></span>.</span></td>
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                      Apply the definitions of midpoint, segment bisector, and angle bisector.
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                    <p id="x-ck12-YWY0MzZhYWFhNmJjNDEyM2Q0YjExZWI3ZTA2YmQ2OWQ.-cqn">Here you'll learn what a midpoint, a segment bisector, and a perpendicular bisector are and how to use their properties to solve for unknown values.</p>
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                     <a itemprop="author" href="/search/?q=author:Dan%20Greenberg">Dan Greenberg</a>
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                     <a itemprop="author" href="/search/?q=author:Lori%20Jordan">Lori Jordan</a>
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                    <span class="js_local_date" data-date="2012-02-24T10:29:52-08:00">February 24, 2012</span>
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                    <span class="js_local_date" data-date="2016-08-25T15:13:06-07:00">August 26, 2016</span>
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                       <a class="keyworditem" href="/search/?q=&amp;subject=mathematics" title="mathematics"><span itemprop="keywords"> mathematics </span></a>
                       <a class="keyworditem" href="/search/?q=&amp;subject=geometry" title="Geometry"><span itemprop="keywords"> Geometry </span></a>
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                       <a class="keyworditem" href="/search/?q=tag:MAT.GEO.122.L.1" title="MAT.GEO.122.L.1"><span itemprop="keywords"> MAT.GEO.122.L.1 </span></a>
                       <a class="keyworditem" href="/search/?q=tag:midpoint%20formula" title="midpoint formula"><span itemprop="keywords"> midpoint formula </span></a>
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                      <a title="MAT.GEO.103.01 (Midpoints and Segment Bisectors - Geometry) " href="/search/?q=Midpoints and Segment Bisectors&amp;type=concept" itemprop="keywords" data-encodedid="MAT.GEO.103.01">MAT.GEO.103.01 (Midpoints and Segment Bisectors - Geometry)</a>
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                &lt;textarea maxlength="1000" id="review_comments" class="feedbacktextarea" placeholder="Give feedback about this resource. Why was it helpful or not? Limit to 1000 max characters." rows="4"&gt;&lt;%= comments %&gt;&lt;/textarea&gt;
            &lt;/div&gt;
            &lt;div class="error profane" id="profane_message"&gt;&lt;/div&gt;
        	&lt;div class="divider"&gt;&lt;/div&gt;
        	&lt;div class="feedbackbtn"&gt;
            &lt;!-- &lt;span class="hint_edittitle"&gt;(Comment) &lt;span class="var_comment"&gt;    0 / 1000&lt;/span&gt;&lt;/span&gt; --&gt;
            &lt;span class="inlineblock wait_js block-for-small"&gt;&lt;/span&gt;
            &lt;span class="spacer"&gt;&lt;/span&gt;
            &lt;span class="inlineblock"&gt;&lt;a id="review_submit"   class="button turquoise feedbackbtn_js " href="javascript:void(0);" &gt;Submit Review&lt;/a&gt;&lt;/span&gt;
            &lt;span class="spacer"&gt;&lt;/span&gt;
            &lt;span class="inlineblock"&gt;&lt;a id="review_skip"  class="button dusty-grey feedbackbtn_skip" href="javascript:void(0);"&gt;Skip Review&lt;/a&gt;&lt;/span&gt;
             &lt;% if (score == 1 || score == -1) { %&gt;
            &lt;span class="inlineblock"&gt;&lt;a class="feedbackbtn_remove"   href="javascript:void(0);"&gt;Remove&lt;/a&gt;&lt;/span&gt;
            &lt;% } else{ %&gt;
            &lt;span class="inlineblock"&gt;&lt;a class="feedbackbtn_cancel"   href="javascript:void(0);"&gt;Cancel&lt;/a&gt;&lt;/span&gt;
            &lt;% } %&gt;
        &lt;/div&gt;
   	 &lt;/div&gt;
    &lt;/div&gt;
</script>
   <!--
Template used for showing the inappropriate comment popup
-->
   <script type="text/x-template" id="commentInappropriateTemplate">
	&lt;div class="flag-comment-container" id="flag-comment-container"&gt;
	    &lt;div class="flag-comment-options"&gt;
	        &lt;span&gt;Does this post seem offensive or disrespectful?&lt;/span&gt;
	        &lt;div class="flag-diff"&gt;&lt;/div&gt;
	        &lt;a class="flag-link" id="inappropriate"&gt;Flag it as inappropriate&lt;/a&gt;
	        &lt;div class="flag-diff"&gt;&lt;/div&gt;
	    &lt;/div&gt;

	    &lt;!--&lt;div class="flag-comment-as-inappropriate hidden"&gt;
	        You flagged this comments as &lt;b&gt;inappropriate&lt;/b&gt;
	        &lt;br&gt;
	        &lt;br&gt;
	        Thanks for bringing this to our attention.
	    &lt;/div&gt;--&gt;
	&lt;/div&gt;
</script>
   <div id="ck12_template_resource_row">
    <div class="resource_row js_resource_row">
     <div class="js_row_info info">
      <div class="resourceicons">
       <span class="attachment_icon imgwrap"></span>
       <span class="noimage js_noimage hide"></span>
       <span class="js_publishstate_icon" title="Private"></span>
      </div>
      <div class="resourcename">
       <div class="resourcenamelinkwrap">
        <a class="break-word js_resource_link" target="_blank" href="javscript:return false;"></a>
       </div>
      </div>
      <div class="clear"></div>
     </div>
     <div class="js_row_actions actions">
      <a href="#" class="js_resource_remove remove" data-resource-id="" data-resource-revision-id="" data-artifact-id="" data-artifact-revision-id=""> <span class="icon-close"></span> </a>
      <a href="#" class="js_resource_edit myresource_edit"><span class="icon-edit"></span></a>
      <span class="hide js_resource_description"></span>
      <a href="#" class="js_resource_public_toggle public_toggle hide" data-txtmakepublic="Make Public" data-txtmakeprivate="Make Private" data-ispublic="false"> Make Public </a>
     </div>
     <div class="js_row_actions loading">
      <div class="progressbar"></div>
     </div>
     <div class="js_row_actions_failed failed hide">
      <div class="message_resources warning">
       <span class="imgwrap"></span>
       <a href="javascript:void(0);" class="js_failed_link" title="Click to see error message">Upload Failed</a>
       <a href="javascript:void(0);" class="js_failed_remove remove" title="Remove Resouce"></a>
      </div>
     </div>
     <div class="clear"></div>
     <div class="divider spacetop"></div>
    </div>
   </div>
   <!--
	Templates for groups sharing
-->
   <!--
	Template used for sharing via groups
-->
   <script type="text/x-template" id="shareGroupsTemplate">
&lt;% if (data.onLoad) { %&gt;
	&lt;div class="row collapse grp-heading"&gt;
		&lt;i class="icon-groups grp-assignment for-group-mobile"&gt;&lt;/i&gt;
		&lt;strong&gt;Share with Classes &lt;/strong&gt;
		&lt;span class="close-modal for-group-mobile"&gt;✕&lt;/span&gt;
		&lt;span class="close-modal" id="close-modal-studyguide-desktops" style="float: right; display: none;"&gt;X&lt;/span&gt;
	&lt;/div&gt;
	&lt;% if (!(data.groups instanceof Array) || data.groups.length == 0) { %&gt;
		&lt;!-- No groups message --&gt;
		&lt;div class="row collapse no_groups_container"&gt;
			&lt;div class="no_groups_text"&gt; You don't own any class.  Please create a class to share this resource.&lt;/div&gt;
			&lt;div class="text-center"&gt;
				&lt;a class="button tangerine standard createnewgroup" title="Create a Class" target="_blank" href="/my/dashboard-new/groups/#create"&gt;Create a Class&lt;/a&gt;
			&lt;/div&gt;
		&lt;/div&gt;
	&lt;% } else { %&gt;
        &lt;div id="shareGroupsContainer" class="relative shareGroupsContainer"&gt;
            &lt;div class="modality-title for-group-mobile"&gt;&lt;/div&gt;
            &lt;div class="custom-box checkbox all-groups for-group-mobile"&gt;
                &lt;input type="checkbox" id="all-grps" name="all-grps"&gt;
                &lt;label id="all-checkbox" for="all-grps" class="checkbox grps-label"&gt;
                    &lt;span&gt;
                        &lt;i class="icon-validated"&gt;&lt;/i&gt;
                    &lt;/span&gt;
                    &lt;i class="all-lbl-txt"&gt;Classes&lt;/i&gt;
                &lt;/label&gt;
            &lt;/div&gt;
            &lt;div id="none_selected_error" class="none_selected_error hide"&gt;Please select at least one class to share.&lt;/div&gt;
            &lt;% _.each(data.groups,function(group) { %&gt;
            &lt;div class="custom-box groups_checkbox"&gt;
                &lt;input type="checkbox" id="&lt;%= group.id %&gt;" name="group_checkbox" value="&lt;%= group.id %&gt;" class="js_groups_select"&gt;
                &lt;label class="checkbox" for="&lt;%= group.id %&gt;" title="&lt;%= _.escape(group.name) %&gt;"&gt;&lt;span&gt;&lt;i class="icon-validated"&gt;&lt;/i&gt;&lt;/span&gt;&lt;i&gt;&lt;%= _.escape(_.truncate(group.name, 35)) %&gt;&lt;/i&gt;&lt;/label&gt;
            &lt;/div&gt;
            &lt;% }); %&gt;
        &lt;/div&gt;
        &lt;span class="load-more-groups for-group-mobile" title="Load more classes"&gt;load more&lt;/span&gt;
        &lt;div class="row collapse button_container"&gt;
			&lt;a id="shareGroupFormBtn" class="button tangerine small" title="Share with the selected class(es)"&gt;Share&lt;/a&gt;

			&lt;span class="or-text"&gt;or&lt;/span&gt;
			&lt;a class="createnew" title="Create a New Class" target="_blank" href="/my/dashboard-new/groups/#create"&gt;Create a New Class&lt;/a&gt;
		&lt;/div&gt;
	&lt;% } %&gt;
&lt;% } else { %&gt;
	&lt;% _.each(data.groups,function(group) { %&gt;
		&lt;div class="custom-box groups_checkbox"&gt;
			&lt;input type="checkbox" id="&lt;%= group.id %&gt;" name="group_checkbox" value="&lt;%= group.id %&gt;" class="js_groups_select"&gt;
			&lt;label class="checkbox" for="&lt;%= group.id %&gt;" title="&lt;%= _.escape(group.name) %&gt;"&gt;&lt;span&gt;&lt;i class="icon-validated"&gt;&lt;/i&gt;&lt;/span&gt;&lt;i&gt;&lt;%= _.escape(_.truncate(group.name, 35)) %&gt;&lt;/i&gt;&lt;/label&gt;
		&lt;/div&gt;
	&lt;% }); %&gt;
&lt;% } %&gt;
</script>
   <!--
	Template used for loading groups.
-->
   <script type="text/x-template" id="shareGroupsLoadingTemplate">
	&lt;div class="text-center"&gt;
		&lt;img src="//img1.ck12.org/media/build-20180828193300/images/icon_loading.gif" /&gt;
	&lt;/div&gt;
</script>
   <div id="js_modality_published_dialog" data-dialogtitle="Modality Published">
   </div>
   <!-- Standards correlation dialog -->
   <script type="text/javascript" src="//maps.googleapis.com/maps/api/js?libraries=places&amp;key=AIzaSyDVTA7d25Puy-_W52IfMDdflx93pWYDJB8"></script>
   <script src="//st2.ck12.org/media/build-20180828193300/dexterjs/dexterjs.min.js"></script>
   <script src="//st1.ck12.org/media/build-20180828193300/common/js/utils/globals.js"></script>
   <script data-main="//st3.ck12.org/media/build-20180828193300/js/main" src="//st3.ck12.org/media/build-20180828193300/common/vendor/requirejs/require.js"></script>
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    function getCookie(c_name){
        var i,x,y,ARRcookies=document.cookie.split(";");
        for (i=0;i&lt;ARRcookies.length;i++){
          x=ARRcookies[i].substr(0,ARRcookies[i].indexOf("="));
          y=ARRcookies[i].substr(ARRcookies[i].indexOf("=")+1);
          x=x.replace(/^\s+|\s+$/g,"");
          if (x==c_name) {
            return unescape(y);
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    }

    function getRandomString(length) {
        var chars = '0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ', value = '';
        for (var i = length; i &gt; 0; --i) {
            value += chars[Math.round(Math.random() * (chars.length - 1))];
        }
        return value;
    }

    // login user role is c.user.role for guest role is in cookies.
    var flxweb_role;
    var flxweb_roles = [];
    var isEditAllowed = false;
    var webroot_url = 'https://www.ck12.org/';
    var editprofile_url = 'https://www.ck12.org/auth/profile?returnTo=https%3A%2F%2Fwww.ck12.org%2Faccount%2Fsettings%2F&amp;appHome=https%3A%2F%2Fwww.ck12.org%2F';
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    _ck12 = dexterjs;

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    function adsLogPageTime() {
        var endTime = new Date().getTime();
        var timeOnPage = Math.round((endTime - window.adsPageStartTime)/1000);
        //pageType, URL, duration (in seconds)
        var payload = {
            'URL' : document.location.href,
            'duration': timeOnPage
        };
        if(window.adsPage) {
            payload['pageType'] = window.adsPage;
        } else if (window.pageType) {
            payload['pageType'] = window.pageType;
        }

        if (window.adsSubject) {
            payload['subject'] = window.adsSubject;
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        if (window.adsBranch) {
            payload['branch'] = window.adsBranch;
        }
        if (window.adsContextEid) {
            payload['context_eid'] = window.adsContextEid;
        }
        if (window.artifactID) {
            payload['artifactID'] = window.artifactID;
        }

        _ck12.logEvent('FBS_TIMESPENT', payload);
     }

    function adsOnload() {
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        startPageTimer();
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    function adsOnBeforeUnload() {
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        var time = now.getTime();
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        now.setTime(expireTime);
        var activeCookie = getCookie('flxweb_user_active');
        if(activeCookie){
            value = activeCookie;
        }else{
            value = getRandomString(25);
            payload = {
                'sessionID': value
            };
            _ck12.logEvent('FBS_ACTIVE_USER', payload);
        }
        document.cookie =
            'flxweb_user_active=' + value +
            '; expires=' + now.toUTCString() +
            '; path=/';
    }

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    addEvent(window, "load", adsOnload);
    addEvent(window, "beforeunload", adsOnBeforeUnload);

&lt;!--Browser Cookie detection start --&gt;
var cookieEnabled = (navigator.cookieEnabled) ? true : false;
if (typeof navigator.cookieEnabled == "undefined" &amp;&amp; !cookieEnabled)
{
    var cookieName = "CK12-Cookie-Test";
    document.cookie= cookieName;
    cookieEnabled = (document.cookie.indexOf(cookieName) != -1) ? true : false;
    if (cookieEnabled){
        document.cookie = cookieName + "=; expires=Thu, 01 Jan 1970 00:00:01 GMT;";
    }
}
if (!cookieEnabled){
    cookieBar('block');
}

function cookieBar(displayValue){
    var cookies_msg_container = document.getElementById("nocookies");
    if (cookies_msg_container != null){
        cookies_msg_container.style.display = displayValue;
    }
}

</script>
   <!--Browser Cookie detection end -->
   <script language="javascript" type="text/javascript">
	require(['main'],function(){
        require(['common/utils/user'], function(User) {
            var isPublished = 'True';
            User.getUser().done(function (user) {
                if (isPublished.toLowerCase() != 'true') {
                    user.isLoggedIn(true);
                }
            });
            require(['flxweb.details.actions']);
        });
	});
</script>
   <script language="JavaScript" type="text/x-mathjax-config;executed=true">
	MathJax.Hub.Config({
		extensions: ["tex2jax.js","TeX/AMSmath.js","TeX/AMSsymbols.js"],
		tex2jax: {
			inlineMath: [['@$','@$']],
			displayMath: [['@$$','@$$']],
			skipTags: ["script","noscript","style","textarea","code"]
		},
		showMathMenu : false,
		jax: ["input/TeX","output/HTML-CSS"],
		messageStyle: "none",
		TeX: {
			extensions: ["cancel.js", "color.js", "autoload-all.js"]
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   <script language="JavaScript" type="text/javascript" src="//d3lkjnwva6z405.cloudfront.net/MathJax-2.7.2/MathJax.js"></script>
   <script type="text/javascript">
    //mod common
    require(['main'],function(){
        require(['js/flxweb.lms.banner'],function(lmsBanner){
            lmsBanner('concept');
	});
    });
</script>
   <script language="JavaScript" type="text/javascript">
	var artifactID = '7493';
	var artifactRevisionID = '4325976';
	var standardBoards = "";
	require(['main'],function(){
		require(['flxweb.details.modality.read']);
        var isPublished = 'True';
        require(['common/utils/user'], function(User) {
            User.getUser().done(function (user) {
                if (user.isLoggedIn(false)) {
                    if (user.userInfoDetail.id == $('.book_unpublished').data('ownerid')) {
                        if (isPublished.toLowerCase() != 'true') {
                            $('.book_unpublished').removeClass('hide');
                        } else {
                            $('.book_published').removeClass('hide');
                        }
                    }
                }
            });
        });
	});
</script>
   <script language="JavaScript" type="text/javascript">
			var modality_type = "lesson"
			require(['main'],function(){
					require(['flxweb.modalities.common']);
			});
	</script>
   <script language="JavaScript" type="text/javascript">
		require(['main'],function(){
			require(['jquery','common/utils/practice_badge'],function($, badge){
				$(document).ready(function() {
					$('.modalitysidebar li a.dropdowntop').click(function() {
						var $dropdown = $(this).next('.dropdown');
						if ($dropdown.is(':hidden')) {
							$dropdown.slideDown('slow');
						}
						else {
							$dropdown.hide('slow');
						}
					});

					function checkForcedSignIn(){
						var selectedUrlconfig = $.cookie('assessmentFrameUrlConfig'),
							$menuElement = null;

						if(selectedUrlconfig &amp;&amp; window.ck12_signed_in){
							$.removeCookie("assessmentFrameUrlConfig");
							window.location.href = selectedUrlconfig;
						}
				 	}

					checkForcedSignIn();
					if ('js_collection_data' in window)
					{
						var jsCollectionData = window.js_collection_data

						//init practice badge if practice available for concept
						 badge &amp;&amp; badge.init({
								"encodedId": window.js_modality_data.domain.encodedID,
								"handle": window.js_modality_data.domain.handle,
								"conceptCollectionHandle": jsCollectionData.collection.descendantCollection.handle,
								"collectionCreatorID": jsCollectionData.collection.creatorID
						 });
					}
					else
					{
						 badge &amp;&amp; badge.init({
								"encodedId": window.js_modality_data.domain.encodedID,
								"handle": window.js_modality_data.domain.handle,
						 });
					}
				});
			});
		});

		/*
		require(['main'],function(){
			require(['jquery'],function($){
				$(document).ready(function(){

				});
			});
		});*/

	</script>
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    if(window.lmsContext === 'lti-app'){
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    }
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