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<h2 class="wp-block-heading">Key Concepts</h2>



<ul class="wp-block-list">
<li>Median and centroid.</li>



<li>Altitudes and orthocenter.</li>



<li>Special case of isosceles triangle.</li>
</ul>



<h2 class="wp-block-heading">Median and Centroid&nbsp;</h2>



<h3 class="wp-block-heading">Median</h3>



<p>A segment from a vertex to the <a href="https://www.turito.com/learn/math/midpoint-and-distance-formulas-grade-9">midpoint</a> of the opposite side is called the median of a triangle.  </p>



<p>A triangle has three vertices and three medians.&nbsp;</p>



<p>The three medians of a triangle are concurrent.&nbsp;</p>



<h3 class="wp-block-heading">Centroid</h3>



<p>The point of intersection of medians is called the centroid.&nbsp;</p>



<p>At a particular point, the triangle will balance, this point is called the balancing point.&nbsp;</p><div class="fillWidthInMob"><a loading='lazy' class="showAdInAllRes"  data-toggle="modal" data-target=#homeLeadForm data-heading="Get Expert Guidance" href="#"><img class="img-fluid" alt="parallel" src=https://a.storyblok.com/f/128066/1100x230/00b28b4876/11-applying-for-collages_leaderboard-1100x230.jpg /></a></div>



<p>This balancing point is also called the centroid.&nbsp;</p>



<h3 class="wp-block-heading">Concurrency of Medians of a Triangle&nbsp;</h3>



<h4 class="wp-block-heading">Theorem</h4>



<p>The medians of a triangle intersect at a point that is two-thirds the distance from each vertex to the midpoint of the opposite side.&nbsp;</p>



<p>The medians of ∆ABC intersect at point O, then CO = <img loading="lazy" decoding="async" width="6" height="27" src="">&nbsp;CD,</p>



<p>AO = 2/3&nbsp;AE, BO = 2/3&nbsp;BF</p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2505.png" alt="Concurrency of Medians of a Triangle" class="wp-image-18148" style="width:279px;height:237px" width="279" height="237" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2505.png 613w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2505-300x254.png 300w" sizes="auto, (max-width: 279px) 100vw, 279px" /></figure></div>


<p><strong>Note:</strong>&nbsp;</p><div class="fillWidthInMob"><a loading='lazy' class="showAdInAllRes"  data-toggle="modal" data-target=#homeLeadForm data-heading="Get Expert Guidance" href="#"><img class="img-fluid" alt="parallel" src=https://a.storyblok.com/f/128066/1100x230/00b28b4876/11-applying-for-collages_leaderboard-1100x230.jpg /></a></div>



<p>The medians of a triangle intersect at a point, at which the median is divided in a 2: 1 ratio from the vertex to the midpoint of the opposite side.&nbsp;&nbsp;&nbsp;</p>



<p><strong>Example 1:&nbsp;</strong></p>



<p>If in triangle ABC, G is the centroid, if AG = 11 units, find GD, AD.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2485.png" alt="Example 1: " class="wp-image-18121" style="width:294px;height:206px" width="294" height="206" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2485.png 740w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2485-300x210.png 300w" sizes="auto, (max-width: 294px) 100vw, 294px" /></figure></div>


<p>Solution:&nbsp;</p>



<p>AG = 2/3 AD&nbsp;</p>



<p>11 =  2/3 AD&nbsp;&nbsp;</p>



<p>AD = 33/2</p>



<p>AD = 16.5</p>



<p>GD = AD – AG</p>



<p>= 16.5 – 11</p>



<p>= 5.5 units</p>



<p>Therefore, AD = 16.5 units, GD = 5.5 units</p>



<p><strong>Example 2:&nbsp;</strong></p>



<p>The vertices of ∆ABC are A (1, 3), B (3, 5), and C (5, 4). Which ordered pair gives the coordinates of the centroid G of ∆ABC?&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="292" height="270" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2486.png" alt="Example 2: " class="wp-image-18122"/></figure></div>


<p>Solution:&nbsp;</p>



<p>The vertices of ∆ABC are A (1, 3), B (3, 5), and C (5, 4)&nbsp;</p>



<p>Draw the graph and plot the midpoint of AC as D&nbsp;</p>



<p>The mid-point of AC&nbsp;=(&nbsp;1+5/2 , 3+4/2&nbsp;)</p>



<p>D=(3,3.5)</p>



<p>The centroid is at a two-third distance from each vertex to the midpoint of the opposite side.&nbsp;&nbsp;</p>



<p>The distance from vertex B (3, 5) to D (3, 3.5) is 5 – 3.5 = 1.5 units.&nbsp;&nbsp;</p>



<p>centroid = 2/3 * 1.5=1&nbsp;</p>



<p>The coordinates of the centroid G are (3, 5 – 1), or (3, 4).&nbsp;</p>



<h3 class="wp-block-heading">Activity</h3>



<p>How to find a balance point (median) of a triangle?&nbsp;</p>



<p>Balancing point:&nbsp;</p>



<p>Step 1:&nbsp;</p>



<p>Take a cardboard and cut it in the shape of a triangle.&nbsp;&nbsp;</p>



<p>Step 2:&nbsp;</p>



<p>Balance the triangle on the eraser end of a pencil.&nbsp;</p>



<p>Step 3:&nbsp;</p>



<p>Mark the point as a balancing point with the marker as B.&nbsp;</p>



<p>Median:&nbsp;</p>



<p>Step 1:&nbsp;</p>



<p>Draw the triangle on the triangle cut out.&nbsp;</p>



<p>Step 2:&nbsp;</p>



<p>Using a ruler, find the midpoints of the triangle.&nbsp;</p>



<p>Step 3:&nbsp;</p>



<p>Draw a segment from the midpoint to the opposite vertex, and point at the intersection point as M.&nbsp;</p>



<p>What do you observe by B and M?&nbsp;</p>



<h2 class="wp-block-heading">Altitudes and Orthocenter&nbsp;</h2>



<h3 class="wp-block-heading">Altitudes&nbsp;</h3>



<p>The altitude of a triangle is the segment that is perpendicular to the opposite side from the vertex&nbsp; or to the line which has an opposite side.&nbsp;</p>



<figure class="wp-block-image size-large"><img loading="lazy" decoding="async" width="1024" height="326" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2508-1024x326.png" alt="Altitudes " class="wp-image-18159" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2508-1024x326.png 1024w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2508-300x95.png 300w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2508-768x244.png 768w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2508-1536x489.png 1536w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2508.png 1600w" sizes="auto, (max-width: 1024px) 100vw, 1024px" /></figure>



<h3 class="wp-block-heading">Concurrency of Altitudes in a Triangle&nbsp;</h3>



<h3 class="wp-block-heading">Theorem</h3>



<p>The lines that have the altitudes of a triangle are concurrent.&nbsp;</p>



<p>AF, BD, and CE are concurrent at point O.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2488.png" alt="Concurrency of Altitudes in a Triangle " class="wp-image-18125" style="width:348px;height:242px" width="348" height="242" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2488.png 747w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2488-300x209.png 300w" sizes="auto, (max-width: 348px) 100vw, 348px" /></figure></div>


<h3 class="wp-block-heading">Orthocenter&nbsp;</h3>



<p>The point at which the lines containing three altitudes of the triangle meet is called the orthocenter or the point of concurrency of the altitudes.&nbsp;&nbsp;</p>



<p><strong>Example:&nbsp;</strong></p>



<h3 class="wp-block-heading">Orthocenter of a triangle&nbsp;</h3>



<p>The orthocenter is the point where all three altitudes of the triangle intersect&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="690" height="227" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2489.png" alt="Orthocenter of a triangle " class="wp-image-18126" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2489.png 690w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2489-300x99.png 300w" sizes="auto, (max-width: 690px) 100vw, 690px" /></figure></div>


<h3 class="wp-block-heading">Special Case&nbsp;</h3>



<h4 class="wp-block-heading"><strong>Isosceles Triangle</strong></h4>



<p>In an isosceles triangle, angle bisector, perpendicular bisector, median, and altitude from vertex point to the opposite side, all the segments are the same.&nbsp;</p>



<p><strong>Remark:</strong>&nbsp;</p>



<p>This is true even in an equilateral triangle with special segments.&nbsp;</p>



<h4 class="wp-block-heading">Property of Isosceles Triangle</h4>



<p><strong>Property</strong></p>



<p>In an isosceles triangle, prove that the median to the base is an altitude.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="194" height="227" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2487.png" alt="Property of Isosceles Triangle" class="wp-image-18123"/></figure></div>


<p><strong>Proof:&nbsp;</strong></p>



<p>Given that ∆ABC is an isosceles triangle, AD is median to the base BC.&nbsp;</p>



<p>Now, we need to prove AD is the altitude to ∆ABC&nbsp;&nbsp;</p>



<p>AC and AB are isosceles in ∆ABC are congruent.&nbsp;</p>



<p>CD = BD, as AD is median&nbsp;</p>



<p>Also, BD = BD&nbsp;</p>



<p>Now ∆ADC ≅&nbsp;∆ABD by SSS congruence postulate,&nbsp;</p>



<p>∠ADC ≅&nbsp;∠ADB are also congruent angles, as there are in congruent triangles.&nbsp;</p>



<p>∠ADC and ∠ADB are linear pairs of angles, AD intersects BC to form a linear pair of angles.&nbsp;</p>



<p>So, AD ⊥ BC, AD is the altitude of ∆ABC.&nbsp;</p>



<p>Hence proved.&nbsp;</p>



<h3 class="wp-block-heading">Activity</h3>



<p>Let us do this activity to understand the difference between perpendicular bisectors, angular bisectors, medians, and altitudes.&nbsp;</p>



<p>Step 1:&nbsp;</p>



<p>Take a cardboard and cut four similar triangles from it.&nbsp;</p>



<p>Step 2:&nbsp;</p>



<p>In the first triangle, draw the perpendicular bisectors and find the point of concurrency.&nbsp;&nbsp;</p>



<p>In the second triangle, draw the angular bisectors and find the point of concurrency,&nbsp;</p>



<p>In the third triangle, draw the medians and find the point of concurrency&nbsp;</p>



<p>In the fourth one, draw the altitudes and find the point of concurrency.&nbsp;</p>



<p>Step 3:&nbsp;</p>



<p>Now compare the four points of concurrency and understand.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="376" height="227" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2490.png" alt="Activity" class="wp-image-18127" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2490.png 376w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2490-300x181.png 300w" sizes="auto, (max-width: 376px) 100vw, 376px" /></figure></div>


<p>If AD = 21 cm, Find AG.</p>



<p>If GC= 16 cm, Find CF.</p>



<p>Solution:</p>



<p>AD = 21 cm</p>



<p>AG = 2/3AD&nbsp;</p>



<p>&nbsp;= 2/3 * 21</p>



<p>= 14 cm</p>



<p>GC = 16 cm</p>



<p>CG = 2/3 * CF</p>



<p>16 = 2/3 * CF</p>



<p>CF=24&nbsp;cm</p>



<h2 class="wp-block-heading">Exercise</h2>



<ul class="wp-block-list" type="1">
<li>Compare a perpendicular bisector and altitude.</li>



<li>Compare an angle bisector and median.</li>
</ul>



<p>In the given figure, G is the centroid</p>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="284" height="235" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2492.png" alt="In the given figure, G is the centroid" class="wp-image-18131"/></figure></div>


<ul class="wp-block-list" type="1">
<li>Find GE.</li>



<li>Find AB</li>



<li>Find BE.</li>



<li>Find BF.</li>



<li>In the given figure, if GE = 4 units, AD = 12 units, CF = 14 units. Find BG.</li>
</ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="289" height="239" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2493.png" alt="In the given figure, if GE = 4 units, AD = 12 units, CF = 14 units. Find BG." class="wp-image-18132"/></figure></div>


<ul class="wp-block-list">
<li>Find ∆ABD ≡ ∆ACD</li>
</ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="288" height="235" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2494.png" alt="Find ∆ABD ≡ ∆ACD" class="wp-image-18133"/></figure></div>


<ul class="wp-block-list">
<li>In an isosceles triangle BAC&nbsp;find DC.</li>
</ul>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="281" height="222" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2495.png" alt="In an isosceles triangle BAC find DC." class="wp-image-18134"/></figure></div>


<ul class="wp-block-list">
<li>Find ∠BAD.</li>
</ul>



<h3 class="wp-block-heading">Concept Map</h3>



<figure class="wp-block-image size-large"><img loading="lazy" decoding="async" width="1024" height="339" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2496-1024x339.png" alt="Concept Map" class="wp-image-18135" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2496-1024x339.png 1024w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2496-300x99.png 300w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2496-768x255.png 768w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2496.png 1068w" sizes="auto, (max-width: 1024px) 100vw, 1024px" /></figure>



<h3 class="wp-block-heading">What have we learned</h3>



<ul class="wp-block-list">
<li>Median and Centroid.</li>



<li>Altitude and Orthocenter.</li>



<li>Special case in Isosceles Triangle.</li>
</ul>
