(14) Kesen Teoremleri

Kenarortay teoremi:


${\displaystyle 2V_a^2=b^2+c^2-\left(\frac{a^2}{2}\right)}$
${\displaystyle \Rightarrow \cancel{2n^2}=\cancel{n^2}+4+4n+\cancel{n^2}+1-2n-\frac{36}{2}}$
${\displaystyle 2n=18-5}$
${\displaystyle n=\frac{13}{\mathrm{2 }}}$

Birinci yol:

Kosinüs teoremi:

${\displaystyle c^2=a^2-2ab\cos C}$

${\displaystyle \Rightarrow 20=25+10-2\times 5\times \sqrt{10}\cos C}$

${\displaystyle \Rightarrow x^2=4+10-2\times 2\times \sqrt{10}\cos C}$

${\displaystyle =14-2\times 2\times \sqrt{10}\cos C\times \frac{5}{5}}$

${\displaystyle =14-\frac{2}{5}\times 15}$

${\displaystyle =8}$
${\displaystyle \Rightarrow x=2\sqrt{2}}$

İkinci yol:
Stewart Teoremi:


${\displaystyle x^2=\left(\frac{b^{2}m+c^{2}n}{m+n}\right)-mn}$

${\displaystyle \Rightarrow x^2=\left(\frac{10\times 3+20\times 2}{5}\right)-6}$

${\displaystyle =8}$

${\displaystyle \Rightarrow x=2\sqrt{2}}$

Birinci yol:




${\displaystyle p^2+36=q^2+25}$

${\displaystyle q^2+9=r^2+4}$

${\displaystyle r^2+16=p^2+x^2}$

${\displaystyle \Rightarrow 61=29+x^2}$

${\displaystyle \Rightarrow x=4\sqrt{2}}$


İkinci yol:


Carnot teoremi:


${\displaystyle a^2+c^2+e^2=b^2+d^2+f^2}$

${\displaystyle \Rightarrow 4+x^2+25=16+36+9}$

${\displaystyle x=4\sqrt{2}}$

${\displaystyle |3x-21|<4x<3x+15}$

${\displaystyle 3x-21<4x<3x+15}$

${\displaystyle \Rightarrow -21<x<15}$

${\displaystyle 21-3x<4x<3x+15}$

${\displaystyle \Rightarrow 3<x<\frac{6x+15}{7}}$

${\displaystyle \Rightarrow 3<x<15}$