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)

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