Ecuatia unei drepte

Ecuatia unei drepte cand se cunosc 2 puncte A(x_a{}, \:\: y_{a}) si B(x_B{}, \:\: y_{B}) prin care trece

a)clasic

AB :   y-y_{A}=\frac{y_{B}- y_{A}}{x_{B}-x_{A}}\cdot\:\:\: (x-x_{A})

b)cu determinant

AB:         \begin{matrix}<br />
x \quad& y & \quad 1\quad \\<br />
  x_{A}\quad &\quad y_{A} &\quad 1 \\<br />
  x_{B}\quad & \quad y_{B} &\quad 1<br />
 \end{matrix}<br />
=<br />
 \begin{matrix*}<br />
\quad \quad  1<br />
 \end{matrix*}

Ecuatia unei drepte cand se cunoaste panta “m” si un punct A(x_a{}, \:\: y_{a}) prin care trece

d: y - y_{A} = m\cdot(x- x_{A})

Determinarea pantei unei drepte

a) cand se cunosc doua puncte prin care trece dreapta

m_{AB} = \frac{y_{B}-y_{A}}{x_{B} - x_{A}}

b) cand se cunoaste ecuatia dreptei

d: ax + by +c = o

m_{d} = -\frac{a}{b}

Fie doua drepte :

d_{1} \quad : \quad \quad \quad a_{1}x +b_{1}x  + c_{1} = 0

d_{2} \quad : \quad \quad \quad a_{2}x +b_{2}x  + c_{2} = 0

d_{1} \parallel d_{2} \Leftrightarrow md_{1}=md_{2}   sau   \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}

d_{1} \perp  d_{2}\Leftrightarrow  md_{1} \cdot md_{2}=-1  sau  a_{1} \cdot a_{2} + b_{1} \cdot b_{2} = 0

Aria \triangle ABC cand se cunosc coordonatele varfurilor A(x_a{}, \:\: y_{a}), B(x_B{}, \:\: y_{B}) si C(x_C{}, \:\: y_{C})

A = \frac{1}{2} \qquad \cdot \qquad |\Delta |
unde  \Delta = \qquad \begin{matrix} x_{A} \quad& y_{A} & \quad 1\quad \\ x_{B}\quad &\quad y_{B} &\quad 1 \\ x_{C}\quad & \quad y_{C} &\quad 1 \end{matrix}
Trei puncte sunt coliniare \Leftrightarrow \Delta = 0

Coordonatele centrului de greutate al \triangle ABC
x_{G}} = \frac{x_{A}+x_{B}+x_{C} }{3}

y_{G}} = \frac{y_{A}+y_{B}+y_{C} }{3}

=> G (x_{G}; \qquad y_{G})

Distanta de la un punct A(x_a{}, \:\: y_{a}) la o dreapta d: \qquad \qquad ax + by + c = 0
{d ( A, d_{1}) = \frac{|a \cdot x_{A}+b \cdot y_{A} + c|}{\sqrt[]{ a^{{2}} + b^{{2}}}
Un patrulater ABCD este paralelogram daca
x_{A} + x_{C} = x_{B} + x_{D}
y_{A} + y_{C} = y_{B} + y_{D}

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