Construct 8

Construct an analytical geometry problem where it is asked to find the vertices of a triangle ABC:
the vertices of this triangle must be the points A (1,7) B (-5,1) C (5, -11). the said problem should be used the concepts of: distance from a point to a line, ratio of division of a segment by a point and angle between two lines.

a) find the length of all sides of a given triangle ABC

Correct answer:

a =  15.6205
b =  18.4391
c =  8.4853

Step-by-step explanation:

$$\begin{gathered} A=(1,7) \\ B=(-5,1) \\ C=(5,-11) \\ a=\mathrm{\mid }BC\mathrm{\mid } \\ a=\sqrt{(C_x-B_x{)}^2+(C_y-B_y{)}^2}=\sqrt{(5-(-5){)}^2+((-11)-1{)}^2}=2\sqrt{61}=15.6205 \end{gathered}$$

$$\begin{gathered} b=\mathrm{\mid }AC\mathrm{\mid } \\ b=\sqrt{(C_x-A_x{)}^2+(C_y-A_y{)}^2}=\sqrt{(5-1{)}^2+((-11)-7{)}^2}=2\sqrt{85}=18.4391 \end{gathered}$$

$$\begin{gathered} b=\mathrm{\mid }AB\mathrm{\mid } \\ c=\sqrt{(B_x-A_x{)}^2+(B_y-A_y{)}^2}=\sqrt{((-5)-1{)}^2+(1-7{)}^2}=6\sqrt{2}=8.4853 \end{gathered}$$

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