Points and vectors

1103030704

Level: 
A
We are given points \( A = [2;1] \), \( B = [4;-1] \), and \( T = [6;2] \), where point \( T \) is the centroid of triangle \( ABC \). Find the length of the median of triangle \( ABC \) to side \( AC \).
\( |t_b|=\frac{\sqrt{117}}2 \)
\( |t_b|=\frac{\sqrt{45}}2 \)
\( |t_b|=\frac{\sqrt{153}}2 \)
\( |t_b|=\sqrt{117} \)

1103030705

Level: 
A
Let there be a triangle KLM and vectors \( \overrightarrow{a} \), \( \overrightarrow{c} \) in the coordinate system. Triangle \( KLM \) and vectors \( \overrightarrow{a} \), \( \overrightarrow{c} \) are given in the coordinate system shown in the picture. Point T is the centroid of the triangle KLM. Express vector \( \overrightarrow{x} \), where \( \overrightarrow{x}=\overrightarrow{KT} \) as a linear combination of \( \overrightarrow{a} \) and \( \overrightarrow{c} \) and evaluate \( \left|\overrightarrow{x}\right| \).
\( \overrightarrow{x}=\frac13 \overrightarrow{a}+\frac13 \overrightarrow{c} \), \( \left|\overrightarrow{x}\right|=5 \)
\( \overrightarrow{x}=\frac23 \overrightarrow{a}+\frac23 \overrightarrow{c} \), \( \left|\overrightarrow{x}\right|=10 \)
\( \overrightarrow{x}=\frac12 \overrightarrow{a}+\frac12 \overrightarrow{c} \), \( \left|\overrightarrow{x}\right|=\frac{15}2 \)
\( \overrightarrow{x}=\frac14 \overrightarrow{a}+\frac14 \overrightarrow{c} \), \( \left|\overrightarrow{x}\right|=\frac{225}{12} \)