Three points

Three points K (-3; 2), L (-1; 4), M (3, -4) are given. Find out:
(a) whether the triangle KLM is right
b) calculate the length of the line to the k side
c) write the coordinates of the vector LM
d) write the directional form of the KM side
e) write the directional form of the axis of the KM side

Result

p =  1
t =  4.4721
x =  4
y =  -8
s = (Correct answer is: s=y=-x-1)

Solution:

$$\begin{gathered} k_x=-3;k_y=2 \\ {l}_x=-1;l_y=4 \\ {m}_x=3;m_y=-4 \\ k=\sqrt{(l_x-m_x{)}^2+(l_y-m_y{)}^2}=\sqrt{((-1)-3{)}^2+(4-(-4){)}^2}=4\sqrt{5}\doteq 8.9443 \\ l=\sqrt{(k_x-m_x{)}^2+(k_y-m_y{)}^2}=\sqrt{((-3)-3{)}^2+(2-(-4){)}^2}=6\sqrt{2}\doteq 8.4853 \\ m=\sqrt{(k_x-l_x{)}^2+(k_y-l_y{)}^2}=\sqrt{((-3)-(-1){)}^2+(2-4{)}^2}=2\sqrt{2}\doteq 2.8284 \\ k>l>m \\ {k}_2=\sqrt{l^2+m^2}=\sqrt{8.4853^2+2.8284^2}=4\sqrt{5}\doteq 8.9443 \\ {k}_2=k \\ \mathrm{\angle }LKM=90^{\circ } \\ p=1 \end{gathered}$$

$t=k\mathrm{/}2=8.9443\mathrm{/}2=2\sqrt{5}=4.4721$

$x=m_x-l_x=3-(-1)=4$

$y=m_y-l_y=(-4)-4=-8$

$$\begin{gathered} f(x)=y=k_x+q \\ k={\displaystyle \frac{k_y-m_y}{k_x-m_x}}={\displaystyle \frac{2-(-4)}{(-3)-3}}=-1 \\ q=m_y-k\cdot m_x=(-4)-(-1)\cdot 3=-1 \\ s=y=-x-1 \end{gathered}$$

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