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<p class="lt-k12-4820">Line segment that joins a vertex and the midpoint of the opposite side of a triangle.</p>
<p class="lt-k12-4820">In a triangle, the line segment that joins a vertex and the midpoint of the opposite side is called a <strong>median</strong>.</p>
<figure><img width="450px" alt="f-d_5142c5476fd2d67c2e72dec68e211aa92cf4a3f6f4cd44ddc279064d+IMAGE_TINY+IMAGE_TINY.png" src="/@api/deki/files/1561/f-d_5142c5476fd2d67c2e72dec68e211aa92cf4a3f6f4cd44ddc279064d%252BIMAGE_TINY%252BIMAGE_TINY.png" />
<figcaption>Figure \(\PageIndex{1}\)</figcaption>
</figure>
<p class="lt-k12-4820">\overline{LO}\) is the median from L\) to the midpoint of \overline{NM}\).</p>
<p class="lt-k12-4820">If you draw all three medians they will intersect at one point called the <strong>centroid</strong>.</p>
<figure><img width="450px" alt="f-d_7680ecc27fedc63c4fd5661e7d16aca3b8ba6fdcdbd759e52e9e95f1+IMAGE_TINY+IMAGE_TINY.png" src="/@api/deki/files/1562/f-d_7680ecc27fedc63c4fd5661e7d16aca3b8ba6fdcdbd759e52e9e95f1%252BIMAGE_TINY%252BIMAGE_TINY.png" />
<figcaption>Figure \(\PageIndex{2}\)</figcaption>
</figure>
<p class="lt-k12-4820">The centroid is the “balancing point” of a triangle. This means that if you were to cut out the triangle, the centroid is its center of gravity so you could balance it there.</p>
<figure><img width="450px" alt="f-d_042f4821fde100ab3d6c12ac37e6f2af5c7ef0131c85d0f1d4709aa8+IMAGE_TINY+IMAGE_TINY.png" src="/@api/deki/files/1563/f-d_042f4821fde100ab3d6c12ac37e6f2af5c7ef0131c85d0f1d4709aa8%252BIMAGE_TINY%252BIMAGE_TINY.png" />
<figcaption>Figure \(\PageIndex{3}\)</figcaption>
</figure>
<p class="lt-k12-4820">The <strong>Median Theorem</strong> states that the medians of a triangle intersect at a point called the centroid that is two-thirds of the distance from the vertices to the midpoint of the opposite sides.</p>
<p class="lt-k12-4820">So if \(G\) is the centroid, then:</p>
<p class="lt-k12-4820">\(AG=\dfrac{2}{3} AD, CG=\dfrac{2}{3} CF,\:EG=\dfrac{2}{3} BE\)</p>
<p class="lt-k12-4820">\(DG=\dfrac{1}{3} AD, FG=\dfrac{1}{3} CF,\:BG=\dfrac{1}{3} BE\)</p>
<p class="lt-k12-4820">\(And\: by \:substitution:DG =\dfrac{1}{2} AG,\:FG=\dfrac{1}{2} CG,\:BG=\dfrac{1}{2} EG\)</p>
<figure><img width="450px" alt="f-d_83f97f564bd89a592bd83f4fdd1e9abe8b02f08a7ac7db40a5a72059+IMAGE_TINY+IMAGE_TINY.png" src="/@api/deki/files/1564/f-d_83f97f564bd89a592bd83f4fdd1e9abe8b02f08a7ac7db40a5a72059%252BIMAGE_TINY%252BIMAGE_TINY.png" />
<figcaption>Figure \(\PageIndex{4}\)</figcaption>
</figure>
<div class="mt-video-widget mt-video-width-55"><img class="mt-media" media="https://www.youtube.com/embed/CUzC6rySeN0" width="450px" aspectratio="56.25%" /></div>
<div class="box-example">
<p class="box-legend lt-k12-4820"><span>Example \(\PageIndex{1}\)</span></p>
<p class="lt-k12-4820">\(B\), \(D\), and \(F\) are the midpoints of each side and \(G\) is the centroid. If \(CG=16\), find \(GF\) and \(CF\).</p>
<figure><img width="450px" alt="f-d_83f97f564bd89a592bd83f4fdd1e9abe8b02f08a7ac7db40a5a72059+IMAGE_TINY+IMAGE_TINY.png" src="/@api/deki/files/1564/f-d_83f97f564bd89a592bd83f4fdd1e9abe8b02f08a7ac7db40a5a72059%252BIMAGE_TINY%252BIMAGE_TINY.png" />
<figcaption>Figure \(\PageIndex{5}\)</figcaption>
</figure>
<p class="lt-k12-4820"><strong>Solution</strong></p>
<p class="lt-k12-4820">Use the Median Theorem.</p>
<p class="mt-align-center lt-k12-4820">\(\begin{align*} CG&=\dfrac{2}{3} CF \\ 16&=\dfrac{2}{3} CF \\ CF&=24.\end{align*}\)</p>
<p class="lt-k12-4820">Therefore, \(GF=8\).</p>
</div>
<div class="box-example">
<p class="box-legend lt-k12-4820"><span>Example \(\PageIndex{2}\)</span></p>
<p class="lt-k12-4820">True or false: The median bisects the side it intersects.</p>
<p class="lt-k12-4820"><strong>Solution</strong></p>
<p class="lt-k12-4820">This statement is true. By definition, a median intersects a side of a triangle at its midpoint. Midpoints divide segments into two equal parts.</p>
</div>
<div class="box-example">
<p class="box-legend lt-k12-4820"><span>Example \(\PageIndex{3}\)</span></p>
<p class="lt-k12-4820">\(I\), \(K\), and \(M\) are midpoints of the sides of \(\Delta HJL\).</p>
<figure><img width="450px" alt="f-d_21e5616ba3b23fb05a6290a3a9c579a121ac9567dc6821bef0612abd+IMAGE_TINY+IMAGE_TINY.png" src="/@api/deki/files/1565/f-d_21e5616ba3b23fb05a6290a3a9c579a121ac9567dc6821bef0612abd%252BIMAGE_TINY%252BIMAGE_TINY.png" />
<figcaption>Figure \(\PageIndex{6}\)</figcaption>
</figure>
<p class="lt-k12-4820"><strong>Solution</strong></p>
<p class="lt-k12-4820">If \(JM=18\), find \(JN\) and \(NM\). If \(HN=14\), find \(NK\) and \(HK\).</p>
<p class="lt-k12-4820">Use the Median Theorem.</p>
<p class="lt-k12-4820">\(JN=\dfrac{2}{3} \cdot 18=12. NM=JM−JN=18−12\). \(NM=6\).</p>
<p class="lt-k12-4820">\(14=\dfrac{2}{3} \cdot HK\)</p>
<p class="lt-k12-4820">\(14\cdot \dfrac{3}{2} =HK=21\). \(NK\) is a third of 21, \(NK=7\).</p>
</div>
<div class="box-example">
<p class="box-legend lt-k12-4820"><span>Example \(\PageIndex{4}\)</span></p>
<p class="lt-k12-4820">H is the centroid of \(\Delta ABC\) and \(DC=5y−16\). Find \(x\) and \(y\).</p>
<figure><img width="450px" alt="f-d_3120a8cbcd0862d22153f88ff4d76d68133944a80de93c1a10e43884+IMAGE_TINY+IMAGE_TINY.png" src="/@api/deki/files/1566/f-d_3120a8cbcd0862d22153f88ff4d76d68133944a80de93c1a10e43884%252BIMAGE_TINY%252BIMAGE_TINY.png" />
<figcaption>Figure \(\PageIndex{7}\)</figcaption>
</figure>
<p class="lt-k12-4820"><strong>Solution</strong></p>
<p class="lt-k12-4820">To solve, use the Median Theorem. Set up and solve equations.</p>
<p class="lt-k12-4820">\(\begin{align*} \dfrac{1}{2} BH=HF &\rightarrow BH=2HF &\qquad HC=\dfrac{2}{3} DC &\rightarrow \dfrac{3}{2} HC=DC \\ 3x+6&=2(2x−1) &\qquad \dfrac{3}{2} (2y+8)&=5y−16\\ 3x+6&=4x−2 &\qquad 3y+12 &=5y−16 \\ 8&=x &\qquad 28&=2y\rightarrow 14=y\end{align*} \) </p>
</div>
<div class="box-example">
<p class="box-legend lt-k12-4820"><span>Example \(\PageIndex{5}\)</span></p>
<p class="lt-k12-4820">\(B\), \(D\), and \(F\) are the midpoints of each side and G is the centroid. If \(BG=5\), find \(GE\) and \(BE\)</p>
<figure><img width="450px" alt="f-d_83f97f564bd89a592bd83f4fdd1e9abe8b02f08a7ac7db40a5a72059+IMAGE_TINY+IMAGE_TINY.png" src="/@api/deki/files/1564/f-d_83f97f564bd89a592bd83f4fdd1e9abe8b02f08a7ac7db40a5a72059%252BIMAGE_TINY%252BIMAGE_TINY.png" />
<figcaption>Figure \(\PageIndex{8}\)</figcaption>
</figure>
<p class="lt-k12-4820"><strong>Solution</strong></p>
<p class="lt-k12-4820">Use the Median Theorem.</p>
<p class="lt-k12-4820">\(\begin{align*} BG&=\dfrac{1}{3} BE \\ 5&=\dfrac{1}{3} BE \\ BE&=15.\end{align*}\)</p>
<p class="lt-k12-4820">Therefore, \(GE=10\).</p>
</div>
<h2 class="lt-k12-4820">Review</h2>
<p class="lt-k12-4820">For questions 1-4, \(B\), \(D\), and \(F\) are the midpoints of each side and \(G\) is the centroid. Find the following lengths.</p>
<figure><img width="450px" alt="f-d_83f97f564bd89a592bd83f4fdd1e9abe8b02f08a7ac7db40a5a72059+IMAGE_TINY+IMAGE_TINY.png" src="/@api/deki/files/1564/f-d_83f97f564bd89a592bd83f4fdd1e9abe8b02f08a7ac7db40a5a72059%252BIMAGE_TINY%252BIMAGE_TINY.png" />
<figcaption>Figure \(\PageIndex{9}\)</figcaption>
</figure>
<ol start="1">
<li class="lt-k12-4820">If \(CG=16\), find \(GF\) and \(CF\)</li>
<li class="lt-k12-4820">If \(AD=30\), find \(AG\) and \(GD\)</li>
<li class="lt-k12-4820">If \(GF=x\), find \(GC\) and \(CF\)</li>
<li class="lt-k12-4820">If \(AG=9x\) and \(GD=5x−1\), find \(x\) and \(AD\).</li>
</ol>
<p class="lt-k12-4820"><strong><em><em>Multi-step Problems</em></em></strong> Find the equation of a median in the x−y\) plane.</p>
<ol start="5">
<li class="lt-k12-4820">Plot \(\Delta ABC:\:A(−6,4)\),\:B(−2,4)\)\:and\:C(6,−4)\)</li>
<li class="lt-k12-4820">Find the midpoint of \(\overline{AC}\). Label it \(D\).</li>
<li class="lt-k12-4820">Find the slope of \(\overline{BD}\).</li>
<li class="lt-k12-4820">Find the equation of \(\overline{BD}\).</li>
<li class="lt-k12-4820">Plot \(\Delta DEF:\: D(−1,5),\:E(1,0),\:F(6,3)\)</li>
<li class="lt-k12-4820">Find the midpoint of \(\overline{EF}\). Label it \(G\).</li>
<li class="lt-k12-4820">Find the slope of \(\overline{DG}\).</li>
<li class="lt-k12-4820">Find the equation of \\(overline{DG}\).</li>
</ol>
<p class="lt-k12-4820">Determine whether the following statement is true or false.</p>
<ol start="13">
<li class="lt-k12-4820">The centroid is the balancing point of a triangle.</li>
</ol>
<h2 class="lt-k12-4820">Review (Answers)</h2>
<p class="lt-k12-4820">To see the Review answers, open this <a href="http://www.ck12.org/flx/show/answer%20key/Answer-Key_CK-12-Chapter-05-Basic-Geometry-Concepts.pdf">PDF file</a> and look for section 5.4. </p>
<h2 class="lt-k12-4820">Resources</h2>
<div class="mt-video-widget mt-video-width-55"><img class="mt-media" media="https://www.youtube.com/embed/aaIX1rUdrgs" width="450px" aspectratio="56.25%" /></div>
<h2 class="lt-k12-4820">Vocabulary</h2>
<table class="mt-responsive-table">
<thead>
<tr>
<th>Term</th>
<th>Definition</th>
</tr>
</thead>
<tbody>
<tr>
<td data-th="Term" class="lt-k12-4820"><strong>centroid</strong></td>
<td data-th="Definition" class="lt-k12-4820">The centroid is the point of intersection of the medians in a triangle.</td>
</tr>
<tr>
<td data-th="Term" class="lt-k12-4820"><strong>Median</strong></td>
<td data-th="Definition" class="lt-k12-4820">The median of a triangle is the line segment that connects a vertex to the opposite side's midpoint.</td>
</tr>
</tbody>
</table>
<h2 class="lt-k12-4820">Additional Resources</h2>
<div class="box-interactive">
<p class="box-legend lt-k12-4820"><span>Interactive Element</span></p>
<div class="mt-video-widget mt-video-width-55"><iframe allowfullscreen="allowfullscreen" frameborder="0" height="350px" scrolling="no" src="https://www.ck12.org/assessment/tools/geometry-tool/fullscreen.html?qID=541c83f2da2cfe0fa1d15ed4&aid=1824253&cch=geometry-::-medians&eId=MAT.GEO.407.04&cci=3" width="350px" /></div>
</div>
<p class="lt-k12-4820">Video: The Medians of a Triangle</p>
<p class="lt-k12-4820">Activities: Medians Discussion Questions</p>
<p class="lt-k12-4820">Study Aids: Bisectors, Medians, Altitudes Study Guide</p>
<p class="lt-k12-4820">Real World: Medians</p>
