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

Here's the solution for Q-3.

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

Step 1: Find the length of the median $m_a$ to side $a=8$ cm. Using Apollonius' theorem: $b^2 + c^2 = 2\left(m_a^2 + \left(\frac{a}{2}\right)^2\right)$ $10^2 + 14^2 = 2\left(m_a^2 + \left(\frac{8}{2}\right)^2\right)$ $100 + 196 = 2(m_a^2 + 4^2)$ $296 = 2(m_a^2 + 16)$ $148 = m_a^2 + 16$ $m_a^2 = 148 - 16$ $m_a^2 = 132$ $m_a = \sqrt{132} = \sqrt{4 \times 33} = 2\sqrt{33} \text{ cm}$

Step 2: Find the length of the median $m_b$ to side $b=10$ cm. Using Apollonius' theorem: $a^2 + c^2 = 2\left(m_b^2 + \left(\frac{b}{2}\right)^2\right)$ $8^2 + 14^2 = 2\left(m_b^2 + \left(\frac{10}{2}\right)^2\right)$ $64 + 196 = 2(m_b^2 + 5^2)$ $260 = 2(m_b^2 + 25)$ $130 = m_b^2 + 25$ $m_b^2 = 130 - 25$ $m_b^2 = 105$ $m_b = \sqrt{105} \text{ cm}$

Step 3: Find the length of the median $m_c$ to side $c=14$ cm. Using Apollonius' theorem: $a^2 + b^2 = 2\left(m_c^2 + \left(\frac{c}{2}\right)^2\right)$ $8^2 + 10^2 = 2\left(m_c^2 + \left(\frac{14}{2}\right)^2\right)$ $64 + 100 = 2(m_c^2 + 7^2)$ $164 = 2(m_c^2 + 49)$ $82 = m_c^2 + 49$ $m_c^2 = 82 - 49$ $m_c^2 = 33$ $m_c = \sqrt{33} \text{ cm}$

The lengths of the medians are: $m_a = 2\sqrt{33 cm}$ $m_b = \sqrt{105 cm}$ $m_c = \sqrt{33 cm}$

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