Question

Let $a = BC , b = CA , c = AB$ be the side lengths of a triangle $ABC$, and $m$ be the length of the median through $A$. If $a =8, b - c =2, m =6$, then the nearest integer to $b$ is

• A. 7
• B. 8
• C. 9
• D. 10

Answer 1

Answer: (B)

$m ^{2}=\frac{2 b ^{2}+2 c ^{2}- a ^{2}}{4}$
$\Rightarrow 144+64=2\left[ b ^{2}+( b -2)^{2}\right]$
$\Rightarrow 104=2 b ^{2}-4 b +4$
$\Rightarrow b ^{2}-2 b -50=0$
$\Rightarrow ( b -1)^{2}=51$
$\Rightarrow b =1+\sqrt{51} \in(8,9)$