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<div class="x-ck12-data-objectives">
<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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<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">segment markings</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="/media/test2picture.png" width="320" alt="" /></span></p>
<p id="x-ck12-NWZlYmFkOTUyNWQ4ODBmYjU0YmQ0OWJiYjRmZmZjZTA.-czn">A <strong>midpoint</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="/media/test1picture.png" 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="/media/test3picture.png" 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>Midpoint Formula:</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>segment bisector</strong> cuts a line segment into two congruent parts and passes through the midpoint. A <strong>perpendicular bisector</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="/media/test1picture.png" 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">&#160;</p>
<div id="x-ck12-rzf" itemprop="video" itemscope="" itemtype="http://schema.org/VideoObject"><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"></iframe></div>
<div id="x-ck12-awo" itemprop="video" itemscope="" itemtype="http://schema.org/VideoObject"><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"></iframe></div>
<p id="x-ck12-MmIyZDljNjI4YmRlZWQwYTZhZjc1YTMzYjBlZmQ4YzA.-toy">&#160;</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="/media/test2picture.png" 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="/media/test4picture.png" 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="/media/test2picture.png" 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="/media/test1picture.png" 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="/media/test1picture.png" 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="/media/test1picture.png" 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&#160;<span class="x-ck12-mathEditor" data-mathmethod="inline" data-contenteditable="false" data-edithtml="" data-tex="x." data-math-class="x-ck12-math"></span>&#160;</p>
<ol id="x-ck12-OThlZDYwZGIxYzVmMTMzNTlmYjYxMDYwYTljOThlNDk.-ezm" start="5">
<li><span class="x-ck12-img-inline"><!-- @@author="" --><!-- @@license="" --><img data-flx-url="/flx/show/image/201502261424993995495972_91bc4f075eb4d9fcb864ba8ec74d94a6-201502261424996110636917.png" src="/media/test1picture.png" width="200" alt="" /></span></li>
</ol>
<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>
</ol>
<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>
</ol>
<h3><span>Review (Answers)</span></h3>
<p id="x-ck12-NzdhOTU5YTg1NzEwZjgwMzQ4ZGMxZTZkNjk3OGE3OTE.-ifg">To <span>see</span>&#160;the Review answers, open this <a href="http://www.ck12.org/flx/show/answer%20key/Answer-Key_CK-12-Chapter-01-Basic-Geometry-Concepts.pdf">PDF file</a> and look for section 1.4.&#160;</p>
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