Answer:

so it is not an isosceles triangle

Verified answer

Answer:

[tex]\qquad\qquad\boxed { \sf \: \triangle \: ABC \: is \: not \: an \: isosceles \: triangle \: } \\ \\ [/tex]

Step-by-step explanation:

Given, a triangle whose vertexs are A (5,7) ,B (3,5), C (2,1).

Now, we have to check whether this triangle is isosceles triangle or not.

So, Consider

[tex]\sf \: AB = \sqrt{ {(5 - 3)}^{2} + {(7 - 5)}^{2} } = \sqrt{ {2}^{2} + {2}^{2} } = \sqrt{2( {2}^{2} )} = 2 \sqrt{2} \\ \\ [/tex]

Now, Consider

[tex]\sf \: BC = \sqrt{ {(3 - 2)}^{2} + {(5 - 1)}^{2} } = \sqrt{ {1}^{2} + {4}^{2} } = \sqrt{1 + 16} = \sqrt{17} \\ \\ [/tex]

Now, Consider

[tex]\sf \: AC = \sqrt{ {(5 - 2)}^{2} + {(7 - 1)}^{2} } = \sqrt{ {3}^{2} + {6}^{2} } = \sqrt{9 + 36} = \sqrt{45} = 3 \sqrt{5} \\ \\ [/tex]

So, from above calculations, we concluded that

[tex]\sf \: AB \: \ne \: BC \: \ne \: AC \\ \\ [/tex]

[tex]\sf\implies \sf \: \triangle \: ABC \: is \: not \: an \: isosceles \: triangle \\ \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Formula used :

Distance Formula

Let A(x₁, y₁) and B(x₂, y₂) be two points in the cartesian plane, then distance between A and B is given by

[tex]\begin{gathered}\boxed{\tt{ AB \: = \sqrt{ {(x_{2} - x_{1}) }^{2} + {(y_{2} - y_{1})}^{2} }}} \\ \end{gathered} \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

[tex] { \large{ \sf{Additional\:Information}}}[/tex]

1. Section formula

Let A(x₁, y₁) and B(x₂, y₂) be two points in the cartesian plane and C(x, y) be the point which divides AB internally in the ratio m₁ : m₂, then the coordinates of C is given by

[tex]\begin{gathered} \boxed{\tt{ (x, y) = \bigg(\dfrac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}}, \dfrac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}\bigg)}} \\ \end{gathered} \\ [/tex]

2. Mid-point formula

Let A(x₁, y₁) and B(x₂, y₂) be two points in the coordinate plane and C(x, y) be the mid-point of AB, then the coordinates of C is given by

[tex]\begin{gathered}\boxed{\tt{ (x,y) = \bigg(\dfrac{x_{1}+x_{2}}{2}, \dfrac{y_{1}+y_{2}}{2}\bigg)}} \\ \end{gathered} \\ [/tex]

3. Centroid of a triangle

Centroid of a triangle is defined as the point at which the medians of the triangle meet and is represented by the symbol G.

Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of a triangle and G(x, y) be the centroid of the triangle, then the coordinates of G is given by

[tex]\begin{gathered}\boxed{\tt{ (x, y) = \bigg(\dfrac{x_{1}+x_{2}+x_{3}}{3}, \dfrac{y_{1}+y_{2}+y_{3}}{3}\bigg)}} \\ \end{gathered} \\ [/tex]

4. Area of a triangle

Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of a triangle, then the area of triangle is given by

[tex]\begin{gathered}\boxed{\tt{ Area =\dfrac{1}{2}\bigg|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\bigg|}} \\ \end{gathered} \\ [/tex]