Respuesta :
We hace a triangle defined by the coordinates of its vertices.
We have to classify this triangle.
We can start by calculating the distance between the sides:
[tex]\begin{gathered} d_{RS}=\sqrt{(x_r-x_s)^2+(y_r-y_s)^2} \\ d_{RS}=\sqrt{(-2-8)^2+(-3-2)^2} \\ d_{RS}=\sqrt{(-10)^2+(-5)^2} \\ d_{RS}=\sqrt{100+25} \\ d_{RS}=\sqrt{125} \\ d_{RS}=5\sqrt{5} \end{gathered}[/tex][tex]\begin{gathered} d_{RT}=\sqrt{(x_r-x_t)^2+(y_r-y_t)^2} \\ d_{RT}=\sqrt{(-2-4)^2+(-3-5)^2} \\ d_{RT}=\sqrt{(-6)^2+(-8)^2} \\ d_{RT}=\sqrt{36+64} \\ d_{RT}=\sqrt{100} \\ d_{RT}=10 \end{gathered}[/tex][tex]\begin{gathered} d_{ST}=\sqrt{(x_s-x_t)^2+(y_s-y_t)^2} \\ d_{ST}=\sqrt{(8-4)^2+(2-5)^2} \\ d_{ST}=\sqrt{4^2+(-3)^2} \\ d_{ST}=\sqrt{16+9} \\ d_{ST}=\sqrt{25} \\ d_{ST}=5 \end{gathered}[/tex]The three sides have different length so, in terms of the sides, the triangle can be classified as scalene.
Now we look at the angles: we will check if it is a right angle by verifying if it satisfies the Pythagorean theorem.
We will use the shortest sides, ST and RT as the legs and RS will be the hypotenuse.
We can then write:
[tex]\begin{gathered} RS^2=RT^2+ST^2 \\ (\sqrt{125})^2=10^2+5^2 \\ 125=100+25 \\ 125=125 \end{gathered}[/tex]As the Pythagorean theorem is satisfied, this triangle is a right triangle.
Answer: the triangle can be classified as right and scalene.
