Answer:
The triangle is both an Isosceles triangle and a right triangle.
Step-by-step explanation:
Given the vertices of a triangle.
![$ P_{1} = (- 1, 4) $](https://tex.z-dn.net/?f=%24%20P_%7B1%7D%20%3D%20%28-%201%2C%204%29%20%24)
and
![$ P_{3} = (4, - 5) $](https://tex.z-dn.net/?f=%24%20P_%7B3%7D%20%3D%20%284%2C%20-%205%29%20%24)
We find the distance between all the points to determine the length of each side of the triangle.
Distance between any two points, say,
and
is:
![$ \sqrt{\bigg ( \textbf{x}_{\textbf{2}} \hspace{1mm} \textbf{- x}_{\textbf{1}} \bigg )^{\textbf{2}} \textbf{+} \bigg( \textbf{y}_{\textbf{2}} \hspace{1mm} \textbf{- y}_{\textbf{1}} \bigg)^ {\textbf{2}} $](https://tex.z-dn.net/?f=%24%20%5Csqrt%7B%5Cbigg%20%28%20%5Ctextbf%7Bx%7D_%7B%5Ctextbf%7B2%7D%7D%20%5Chspace%7B1mm%7D%20%5Ctextbf%7B-%20x%7D_%7B%5Ctextbf%7B1%7D%7D%20%5Cbigg%20%29%5E%7B%5Ctextbf%7B2%7D%7D%20%5Ctextbf%7B%2B%7D%20%20%20%5Cbigg%28%20%5Ctextbf%7By%7D_%7B%5Ctextbf%7B2%7D%7D%20%5Chspace%7B1mm%7D%20%5Ctextbf%7B-%20y%7D_%7B%5Ctextbf%7B1%7D%7D%20%5Cbigg%29%5E%20%7B%5Ctextbf%7B2%7D%7D%20%24)
Length between
and
, (Side 1) :
and
![$ (x_2, y_2) = (6, 2) $](https://tex.z-dn.net/?f=%24%20%28x_2%2C%20y_2%29%20%3D%20%286%2C%202%29%20%24)
Distance = ![$ \sqrt{\bigg(6 - (-1) \bigg)^{2} \hspace{1mm} + \hspace{1mm} \bigg( 2 - 4 \bigg )^2 $](https://tex.z-dn.net/?f=%24%20%5Csqrt%7B%5Cbigg%286%20-%20%28-1%29%20%5Cbigg%29%5E%7B2%7D%20%5Chspace%7B1mm%7D%20%2B%20%5Chspace%7B1mm%7D%20%5Cbigg%28%202%20-%204%20%5Cbigg%20%29%5E2%20%24)
![$ = \sqrt{7^2 + 2^2} $](https://tex.z-dn.net/?f=%24%20%3D%20%5Csqrt%7B7%5E2%20%2B%202%5E2%7D%20%24)
![$ = \sqrt{49 + 4} $](https://tex.z-dn.net/?f=%24%20%3D%20%5Csqrt%7B49%20%2B%204%7D%20%24)
Length of Side 1 =
units.
Distance between
and
, (Side 2):
![$ (x_1, y_1) = (-1, 4) $](https://tex.z-dn.net/?f=%24%20%28x_1%2C%20y_1%29%20%3D%20%28-1%2C%204%29%20%24)
![$ (x_2, y_2) = (4, - 5) $](https://tex.z-dn.net/?f=%24%20%28x_2%2C%20y_2%29%20%3D%20%284%2C%20-%205%29%20%24)
Distance = ![$ \sqrt{ \bigg( 4 + 1 \bigg)^2 \hspace{1mm} + \bigg( - 5 - 4 \bigg ) ^2 $](https://tex.z-dn.net/?f=%24%20%5Csqrt%7B%20%5Cbigg%28%204%20%2B%201%20%5Cbigg%29%5E2%20%5Chspace%7B1mm%7D%20%2B%20%5Cbigg%28%20-%205%20-%204%20%5Cbigg%20%29%20%5E2%20%24)
![$ = \sqrt{ 25 + 81 } $](https://tex.z-dn.net/?f=%24%20%3D%20%5Csqrt%7B%2025%20%2B%2081%20%7D%20%24)
![$ = \sqrt{\textbf{106}} $](https://tex.z-dn.net/?f=%24%20%3D%20%5Csqrt%7B%5Ctextbf%7B106%7D%7D%20%24)
Length of Side 2 =
units.
Distance between
and
, Side 3 :
![$ (x_2, y_2) = (6, 2) $](https://tex.z-dn.net/?f=%24%20%28x_2%2C%20y_2%29%20%3D%20%286%2C%202%29%20%24)
Distance = ![$ \sqrt{ 2^2 \hspace{1mm} + \hspace{1mm} 7^2} $](https://tex.z-dn.net/?f=%24%20%5Csqrt%7B%202%5E2%20%5Chspace%7B1mm%7D%20%2B%20%5Chspace%7B1mm%7D%207%5E2%7D%20%24)
![$ = \sqrt{49 + 4} $](https://tex.z-dn.net/?f=%24%20%3D%20%5Csqrt%7B49%20%2B%204%7D%20%24)
![$ = \sqrt{\textbf{53} $](https://tex.z-dn.net/?f=%24%20%3D%20%5Csqrt%7B%5Ctextbf%7B53%7D%20%24)
Length of Side 3 =
units.
Note that the length of Side 1 = Length of Side 3.
That means the triangle is Isosceles.
Also, For a triangle to be right angle triangle, using Pythagoras theorem we have:
(Side 1)² + (Side 3)² = (Side 2)²
![$ \bigg( \sqrt{53} \bigg )^2 \hspace{1mm} + \hspace{1mm} \bigg( \sqrt{53} \bigg)^2 \hspace{1mm} = \hspace{1mm} \bigg ( \sqrt{106} \bigg ) ^2 $](https://tex.z-dn.net/?f=%24%20%5Cbigg%28%20%5Csqrt%7B53%7D%20%5Cbigg%20%29%5E2%20%5Chspace%7B1mm%7D%20%2B%20%5Chspace%7B1mm%7D%20%5Cbigg%28%20%5Csqrt%7B53%7D%20%5Cbigg%29%5E2%20%5Chspace%7B1mm%7D%20%3D%20%5Chspace%7B1mm%7D%20%5Cbigg%20%28%20%5Csqrt%7B106%7D%20%5Cbigg%20%29%20%5E2%20%24)
i.e, 53 + 53 = 106
Hence, the triangle is a right - angled triangle as well.