Question

A-determine o ângulo agudo formado pelas retas seguintes :​

Answer 1

$\boxed{\begin{array}{l}\tt a)~\sf r:4x-3y+5=0\implies m_r=-\dfrac{4}{-3}=\dfrac{4}{3}\\\sf s:3x+4y+2=0\implies m_s=-\dfrac{3}{4}\\\sf m_r\cdot m_s=\dfrac{4}{3}\cdot\bigg(-\dfrac{3}{4}\bigg)\\\sf m_r\cdot m_s=-1\\\sf \theta=90^\circ\\\sf n\tilde ao~h\acute a~\hat angulo~agudo~entre~r~e~s.\end{array}}$

$\boxed{\begin{array}{l}\tt b)~\sf r:x+y-5=0\implies m_r=\dfrac{-1}{1}=-1\\\sf s:y+4=0\implies m_s=0\\\sf tg(\theta)=\bigg|\dfrac{m_s-m_r}{1+m_s\cdot m_r}\bigg|\\\\\sf tg(\theta)=\bigg|\dfrac{1}{1}\bigg|\\\sf tg(\theta)=1\\\sf\theta=arctg(1)\\\sf\theta=45^\circ\end{array}}$

$\boxed{\begin{array}{l}\tt c)~\sf r:5x-y+7=0\implies m_r=-\dfrac{5}{-1}=5\\\sf s:3x+2y=0\implies m_s=-\dfrac{3}{2}=1,5\\\sf tg(\theta)\bigg|\dfrac{1,5-5}{1+1,5\cdot5}\bigg|\\\sf tg(\theta)=\dfrac{3,5\div0,5}{8,5\div0,5}=\dfrac{7}{17}\\\sf \theta=arctg\bigg(\dfrac{7}{17}\bigg)\end{array}}$

$\boxed{\begin{array}{l}\tt d)~\sf r:3x-2y+5=0\implies m_r=-\dfrac{3}{-2}=\dfrac{3}{2}=\dfrac{15}{10}\\\sf s: 4x+5y-2=0\implies m_s=-\dfrac{4}{5}=-\dfrac{8}{10}\\\sf l 1=\dfrac{10}{10}\\\sf tg(\theta)=\bigg|\dfrac{\frac{8}{10}-\frac{15}{10}}{\frac{10}{10}-\frac{8}{10}\cdot\frac{15}{10}}\bigg|\\\\\sf tg(\theta)=\bigg|\dfrac{-\frac{7}{\diagdown\!\!\!\!\!10}}{-\frac{110}{\diagdown\!\!\!\!\!10}}\bigg|\\\\\sf tg(\theta)=\dfrac{7}{110}\\\sf \theta=arctg\bigg(\dfrac{7}{110}\bigg)\end{array}}$