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Find the length of the median m_a to side a=8 cm.

Mathematics

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Answer

ma=233 cmm_a = 2\sqrt{33} \text{ cm}

mb=105 cmm_b = \sqrt{105} \text{ cm}

mc=33 cmm_c = \sqrt{33} \text{ cm}

Step-by-step solution

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Here's the solution for Q-3.

To find the length of the medians of a triangle with sides a=8a=8 cm, b=10b=10 cm, and c=14c=14 cm, we use Apollonius' theorem. Apollonius' theorem states that for a triangle with sides a,b,ca, b, c and a median mam_a to side aa, the relationship is b2+c2=2(ma2+(a2)2)b^2 + c^2 = 2(m_a^2 + (\frac{a}{2})^2). We will apply this formula for each median.

  1. Find the length of the median mam_a to side a=8a=8 cm.

    Using Apollonius' theorem:

    b2+c2=2(ma2+(a2)2)b^2 + c^2 = 2\left(m_a^2 + \left(\frac{a}{2}\right)^2\right) 102+142=2(ma2+(82)2)10^2 + 14^2 = 2\left(m_a^2 + \left(\frac{8}{2}\right)^2\right) 100+196=2(ma2+42)100 + 196 = 2(m_a^2 + 4^2) 296=2(ma2+16)296 = 2(m_a^2 + 16) 148=ma2+16148 = m_a^2 + 16 ma2=148−16m_a^2 = 148 - 16 ma2=132m_a^2 = 132 ma=132=4×33=233 cmm_a = \sqrt{132} = \sqrt{4 \times 33} = 2\sqrt{33} \text{ cm}

  2. Find the length of the median mbm_b to side b=10b=10 cm.

    Using Apollonius' theorem:

    a2+c2=2(mb2+(b2)2)a^2 + c^2 = 2\left(m_b^2 + \left(\frac{b}{2}\right)^2\right) 82+142=2(mb2+(102)2)8^2 + 14^2 = 2\left(m_b^2 + \left(\frac{10}{2}\right)^2\right) 64+196=2(mb2+52)64 + 196 = 2(m_b^2 + 5^2) 260=2(mb2+25)260 = 2(m_b^2 + 25) 130=mb2+25130 = m_b^2 + 25 mb2=130−25m_b^2 = 130 - 25 mb2=105m_b^2 = 105 mb=105 cmm_b = \sqrt{105} \text{ cm}

  3. Find the length of the median mcm_c to side c=14c=14 cm.

    Using Apollonius' theorem:

    a2+b2=2(mc2+(c2)2)a^2 + b^2 = 2\left(m_c^2 + \left(\frac{c}{2}\right)^2\right) 82+102=2(mc2+(142)2)8^2 + 10^2 = 2\left(m_c^2 + \left(\frac{14}{2}\right)^2\right) 64+100=2(mc2+72)64 + 100 = 2(m_c^2 + 7^2) 164=2(mc2+49)164 = 2(m_c^2 + 49) 82=mc2+4982 = m_c^2 + 49 mc2=82−49m_c^2 = 82 - 49 mc2=33m_c^2 = 33 mc=33 cmm_c = \sqrt{33} \text{ cm}

    The lengths of the medians are:
    ma=233 cmm_a = \boxed{2\sqrt{33} \text{ cm}}
    mb=105 cmm_b = \boxed{\sqrt{105} \text{ cm}}
    mc=33 cmm_c = \boxed{\sqrt{33} \text{ cm}}

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