This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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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 + ((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(m_a^2 + ((a)/(2))^2) 10^2 + 14^2 = 2(m_a^2 + ((8)/(2))^2) 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 × 33) = 2sqrt(33) cm Step 2: Find the length of the median m_b to side b=10 cm. Using Apollonius' theorem: a^2 + c^2 = 2(m_b^2 + ((b)/(2))^2) 8^2 + 14^2 = 2(m_b^2 + ((10)/(2))^2) 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) cm Step 3: Find the length of the median m_c to side c=14 cm. Using Apollonius' theorem: a^2 + b^2 = 2(m_c^2 + ((c)/(2))^2) 8^2 + 10^2 = 2(m_c^2 + ((14)/(2))^2) 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) cm The lengths of the medians are: m_a = 2sqrt(33) cm m_b = sqrt(105) cm m_c = sqrt(33) cm Last free one today — make it count tomorrow, or type /upgrade for unlimited.