Answer:
The three given points are the vertices of a right triangle.
Step-by-step explanation:
To determine that the three points are the vertices of a right triangle let us find the distance between each two points
The formula of the distance is [tex]d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2} }[/tex]
∵ [tex]x_{1}[/tex] = 6 and [tex]y_{1}[/tex] = -6
∵ [tex]x_{2}[/tex] = 9 and [tex]y_{2}[/tex] = -6
∴ [tex]d_{1}[/tex] = [tex]\sqrt{(9-6)^{2}+(-6--6)^{2}}=\sqrt{9+0}[/tex]
∴ [tex]d_{1}[/tex] = 3
∵ [tex]x_{1}[/tex] = 6 and [tex]y_{1}[/tex] = -6
∵ [tex]x_{2}[/tex] = 9 and [tex]y_{2}[/tex] = 1
∴ [tex]d_{2}[/tex] = [tex]\sqrt{(9-6)^{2}+(1--6)^{2}}=\sqrt{9+49}[/tex]
∴ [tex]d_{2}[/tex] = [tex]\sqrt{58}[/tex]
∵ [tex]x_{1}[/tex] = 9 and [tex]y_{1}[/tex] = 1
∵ [tex]x_{2}[/tex] = 9 and [tex]y_{2}[/tex] = -6
∴ [tex]d_{3}[/tex] = [tex]\sqrt{(9-9)^{2}+(-6-1)^{2}}=\sqrt{0+49}[/tex]
∴ [tex]d_{3}[/tex] = 7
Let us use the fact that if the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle at the vertex which opposite to the longest side
∵ The longest side is [tex]\sqrt{58}[/tex]
∵ The other two sides are 3 and 7
∵ ( [tex]\sqrt{58}[/tex] )² = 58
∵ (3)² + (7)² = 9 + 49 = 58
∴ ( [tex]\sqrt{58}[/tex] )² = (3)² + (7)²
- By using the fact above
∴ The triangle is a right triangle
The three given points are the vertices of a right triangle.
