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1.
\(\left\{{}\begin{matrix}x_I=\dfrac{x_A+x_B}{2}=-\dfrac{3}{2}\\y_I=\dfrac{y_A+y_B}{2}=1\end{matrix}\right.\) \(\Rightarrow I\left(-\dfrac{3}{2};1\right)\)
\(\left\{{}\begin{matrix}x_G=\dfrac{x_A+x_B+x_C}{3}=0\\y_G=\dfrac{y_A+y_B+y_C}{3}=0\end{matrix}\right.\) \(\Rightarrow G\left(0;0\right)\)
2.
\(\left\{{}\begin{matrix}\overrightarrow{CI}=\left(-\dfrac{9}{2};3\right)\\\overrightarrow{AG}=\left(-2;-3\right)\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}CI=\sqrt{\left(-\dfrac{9}{2}\right)^2+3^2}=\dfrac{3\sqrt{13}}{2}\\AG=\sqrt{\left(-2\right)^2+\left(-3\right)^2}=\sqrt{13}\end{matrix}\right.\)
3.
Gọi \(D\left(x;y\right)\Rightarrow\left\{{}\begin{matrix}\overrightarrow{AB}=\left(-7;-4\right)\\\overrightarrow{DC}=\left(3-x;-2-y\right)\end{matrix}\right.\)
\(ABCD\) là hbh \(\Leftrightarrow\overrightarrow{AB}=\overrightarrow{DC}\)
\(\Leftrightarrow\left\{{}\begin{matrix}-7=3-x\\-4=-2-y\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=10\\y=2\end{matrix}\right.\)
\(\Rightarrow D\left(10;2\right)\)
4. Gọi \(H\left(x;y\right)\Rightarrow\left\{{}\begin{matrix}\overrightarrow{CH}=\left(x-3;y+2\right)\\\overrightarrow{AH}=\left(x-2;y-3\right)\\\overrightarrow{BC}=\left(8;-1\right)\end{matrix}\right.\)
H là trực tâm \(\Leftrightarrow\left\{{}\begin{matrix}AH\perp BC\\CH\perp AB\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}\overrightarrow{AH}.\overrightarrow{BC}=0\\\overrightarrow{CH}.\overrightarrow{AB}=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}8\left(x-2\right)-1\left(y-3\right)=0\\-7\left(x-3\right)-4\left(y+2\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}8x-y=13\\-7x-4y=-13\end{matrix}\right.\) \(\Rightarrow H\left(\dfrac{5}{3};\dfrac{1}{3}\right)\)
a: Tọa độ trọng tâm là:
\(\left\{{}\begin{matrix}x=\dfrac{1+3+5}{3}=3\\y=\dfrac{1+5-1}{3}=\dfrac{5}{3}\end{matrix}\right.\)
b: \(\overrightarrow{BC}=\left(2;-6\right)\)
\(\overrightarrow{AD}=\left(x-1;y-1\right)\)
Để BC//AD và BC=2AD thì 2=2(x-1) và -6=2(y-1)
=>x-1=1 và y-1=-3
=>x=2 và y=-2
a.
\(\left\{{}\begin{matrix}\overrightarrow{AB}=\left(-1;8\right)\\\overrightarrow{AC}=\left(3;6\right)\end{matrix}\right.\) mà \(\dfrac{-1}{3}\ne\dfrac{8}{6}\Rightarrow\overrightarrow{AB}\) và \(\overrightarrow{AC}\) không cùng phương hay A,B,C không thẳng hàng
\(\Rightarrow A,B,C\) là 3 đỉnh của 1 tam giác
b.
Theo công thức trung điểm: \(\left\{{}\begin{matrix}x_I=\dfrac{x_A+x_C}{2}=\dfrac{1+4}{2}=\dfrac{5}{2}\\y_I=\dfrac{y_A+y_C}{2}=\dfrac{-3+3}{2}=0\end{matrix}\right.\)
\(\Rightarrow C\left(\dfrac{5}{2};0\right)\)
Gọi G là trọng tâm tam giác, theo công thức trọng tâm:
\(\left\{{}\begin{matrix}x_G=\dfrac{x_A+x_B+x_C}{3}=\dfrac{1+0+4}{3}=\dfrac{5}{3}\\y_G=\dfrac{y_A+y_B+y_C}{3}=\dfrac{-3+5+3}{3}=\dfrac{5}{3}\\\end{matrix}\right.\) \(\Rightarrow G\left(\dfrac{5}{3};\dfrac{5}{3}\right)\)
c.
Gọi \(D\left(x;y\right)\Rightarrow\overrightarrow{DC}=\left(4-x;3-y\right)\)
ABCD là hình bình hành khi \(\overrightarrow{AB}=\overrightarrow{DC}\)
\(\Rightarrow\left\{{}\begin{matrix}4-x=-1\\3-y=8\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=5\\y=-5\end{matrix}\right.\)
\(\Rightarrow D\left(5;-5\right)\)
a: \(\left\{{}\begin{matrix}x_G=\dfrac{2+4+2}{3}=\dfrac{8}{3}\\y_G=\dfrac{1+0+3}{3}=\dfrac{4}{3}\end{matrix}\right.\)
\(\left\{{}\begin{matrix}x_I=\dfrac{2+4}{2}=3\\y_I=\dfrac{1+0}{2}=\dfrac{1}{2}\end{matrix}\right.\)
\(I\left(\frac{3-11}{2};\frac{2+0}{2}\right)\Rightarrow I\left(-4;1\right)\)
\(G\left(\frac{3+5-11}{3};\frac{2+4+0}{3}\right)\Rightarrow G\left(-1;2\right)\)
\(M\left(-22-5;0-4\right)\Rightarrow M\left(-27;-4\right)\)
\(D\left(3+5--11;2+4-0\right)\Rightarrow D\left(19;6\right)\)