Skip to main content
K12 LibreTexts

4.23: Medians

  • Page ID
    4820
  • Line segment that joins a vertex and the midpoint of the opposite side of a triangle.

    In a triangle, the line segment that joins a vertex and the midpoint of the opposite side is called a median.

    f-d_5142c5476fd2d67c2e72dec68e211aa92cf4a3f6f4cd44ddc279064d+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    \overline{LO}\) is the median from L\) to the midpoint of \overline{NM}\).

    If you draw all three medians they will intersect at one point called the centroid.

    f-d_7680ecc27fedc63c4fd5661e7d16aca3b8ba6fdcdbd759e52e9e95f1+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    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.

    f-d_042f4821fde100ab3d6c12ac37e6f2af5c7ef0131c85d0f1d4709aa8+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    The Median Theorem 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.

    So if \(G\) is the centroid, then:

    \(AG=\dfrac{2}{3} AD, CG=\dfrac{2}{3} CF,\:EG=\dfrac{2}{3} BE\)

    \(DG=\dfrac{1}{3} AD, FG=\dfrac{1}{3} CF,\:BG=\dfrac{1}{3} BE\)

    \(And\: by \:substitution:DG =\dfrac{1}{2} AG,\:FG=\dfrac{1}{2} CG,\:BG=\dfrac{1}{2} EG\)

    f-d_83f97f564bd89a592bd83f4fdd1e9abe8b02f08a7ac7db40a5a72059+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Example \(\PageIndex{1}\)

    \(B\), \(D\), and \(F\) are the midpoints of each side and \(G\) is the centroid. If \(CG=16\), find \(GF\) and \(CF\).

    f-d_83f97f564bd89a592bd83f4fdd1e9abe8b02f08a7ac7db40a5a72059+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    Solution

    Use the Median Theorem.

    \(\begin{align*} CG&=\dfrac{2}{3} CF \\ 16&=\dfrac{2}{3} CF \\ CF&=24.\end{align*}\)

    Therefore, \(GF=8\).

    Example \(\PageIndex{2}\)

    True or false: The median bisects the side it intersects.

    Solution

    This statement is true. By definition, a median intersects a side of a triangle at its midpoint. Midpoints divide segments into two equal parts.

    Example \(\PageIndex{3}\)

    \(I\), \(K\), and \(M\) are midpoints of the sides of \(\Delta HJL\).

    f-d_21e5616ba3b23fb05a6290a3a9c579a121ac9567dc6821bef0612abd+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

    Solution

    If \(JM=18\), find \(JN\) and \(NM\). If \(HN=14\), find \(NK\) and \(HK\).

    Use the Median Theorem.

    \(JN=\dfrac{2}{3} \cdot 18=12. NM=JM−JN=18−12\). \(NM=6\).

    \(14=\dfrac{2}{3} \cdot HK\)

    \(14\cdot \dfrac{3}{2} =HK=21\). \(NK\) is a third of 21, \(NK=7\).

    Example \(\PageIndex{4}\)

    H is the centroid of \(\Delta ABC\) and \(DC=5y−16\). Find \(x\) and \(y\).

    f-d_3120a8cbcd0862d22153f88ff4d76d68133944a80de93c1a10e43884+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{7}\)

    Solution

    To solve, use the Median Theorem. Set up and solve equations.

    \(\begin{align*} \dfrac{1}{2} BH=HF &\rightarrow BH=2HF &\qquad HC=\dfrac{2}{3} DC &\rightarrow \dfrac{3}{2} HC=DC \\ 3x+6&=2(2x−1) &\qquad \dfrac{3}{2} (2y+8)&=5y−16\\ 3x+6&=4x−2 &\qquad 3y+12 &=5y−16 \\ 8&=x &\qquad 28&=2y\rightarrow 14=y\end{align*} \)

    Example \(\PageIndex{5}\)

    \(B\), \(D\), and \(F\) are the midpoints of each side and G is the centroid. If \(BG=5\), find \(GE\) and \(BE\)

    f-d_83f97f564bd89a592bd83f4fdd1e9abe8b02f08a7ac7db40a5a72059+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{8}\)

    Solution

    Use the Median Theorem.

    \(\begin{align*} BG&=\dfrac{1}{3} BE \\ 5&=\dfrac{1}{3} BE \\ BE&=15.\end{align*}\)

    Therefore, \(GE=10\).

    Review

    For questions 1-4, \(B\), \(D\), and \(F\) are the midpoints of each side and \(G\) is the centroid. Find the following lengths.

    f-d_83f97f564bd89a592bd83f4fdd1e9abe8b02f08a7ac7db40a5a72059+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{9}\)
    1. If \(CG=16\), find \(GF\) and \(CF\)
    2. If \(AD=30\), find \(AG\) and \(GD\)
    3. If \(GF=x\), find \(GC\) and \(CF\)
    4. If \(AG=9x\) and \(GD=5x−1\), find \(x\) and \(AD\).

    Multi-step Problems Find the equation of a median in the x−y\) plane.

    1. Plot \(\Delta ABC:\:A(−6,4)\),\:B(−2,4)\)\:and\:C(6,−4)\)
    2. Find the midpoint of \(\overline{AC}\). Label it \(D\).
    3. Find the slope of \(\overline{BD}\).
    4. Find the equation of \(\overline{BD}\).
    5. Plot \(\Delta DEF:\: D(−1,5),\:E(1,0),\:F(6,3)\)
    6. Find the midpoint of \(\overline{EF}\). Label it \(G\).
    7. Find the slope of \(\overline{DG}\).
    8. Find the equation of \\(overline{DG}\).

    Determine whether the following statement is true or false.

    1. The centroid is the balancing point of a triangle.

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 5.4.

    Resources

    Vocabulary

    Term Definition
    centroid The centroid is the point of intersection of the medians in a triangle.
    Median The median of a triangle is the line segment that connects a vertex to the opposite side's midpoint.

    Additional Resources

    Interactive Element

    Video: The Medians of a Triangle

    Activities: Medians Discussion Questions

    Study Aids: Bisectors, Medians, Altitudes Study Guide

    Real World: Medians