In the isosceles triangle ABC with BC being the unequal side of the triangle, and line AD is the median drawn from the vertex A to the side BC. If the length AC = 5 cm and the length of the median is 4 cm. then find the length of BC (in cm).

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Since \(\Delta \)ABC is an isosceles triangle,

Therefore, AB = AC = 5 cm

Apply Apollonius' theorem

= \( {5 }^{2 } \) + \( {5 }^{2 } \) = 2(\( {4 }^{2 } \) + \( {DC }^{2 } \))

= 25 + 25 = 2(16 + \( {DC }^{2 } \))

= 50 = 2(16 + \( {DC }^{2 } \))

= 25 = 16 + \( {DC }^{2 } \)

= \( {DC }^{2 } \) = 25 - 16

= \( {DC }^{2 } \) = 9

= DC = 3 cm

Since D is he median point on the side BC,

Therefore, BD = DC = 3 cm

= BC = BD + CD

= BC = 3 + 3 = 6 cm

Therefore, BC is 6 cm.