Question

In a triangle ABC, AB = 7 cm, AC
9 cm and BC = 14 cm. Find the length of the median drawn from A to BC?
Select one:
a. 8 cm
b. 4 cm
c. 5 cm
d. 6 cm​

Answer 1

$In \: \triangle ABC , AB = 7 \:cm , BC = 14 \:cm \\and \: AC = 9 \: cm$

$AD \: is \: the \: median .$

$BD = \frac{BC}{2} \\= \frac{14}{2} \\= 7 \:cm$

$\underline { \pink { Apollonius \: Theorem: }}$

The sum of the squares of any two sides of any triangle equals twice the square of half the third side , together with twice the square of the median bisecting the third side .

$\boxed { \blue{ AB^{2} + AC^{2} = 2(AD^{2} + BD^{2} ) }}$

$\implies 7^{2} + 9^{2} = 2( AD^{2} + 7^{2} )$

$\implies 49 + 81 = 2( AD^{2} + 49 )$

$\implies 130 = 2( AD^{2} + 49 )$

$\implies \frac{130}{2} = AD^{2} + 49$

$\implies 65 = AD^{2} + 49$

$\implies AD^{2} + 49 = 65$

$\implies AD^{2} = 65 - 49$

$\implies AD^{2} = 16$

$\implies AD= \sqrt{4^{2}}$

$\implies AD = 4 \:cm$

Therefore.,

$Option \: \pink { ( b ) } \: is \: correct .$

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