Question

Name the type of triangle formed by the points \[A( - 5,6)\], \[B( - 4, - 2)\] and \[C(7,5)\].

Answer 1

Hint:
According to the question, you need to calculate distance between the two points using distance formula \[\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \] . Check whether the distance between the two points is equal or not and hence examine the triangle formed by the calculated distance.

Formula used:
Here, we use the Distance Formula = \[\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]

Complete step by step solution:
It is given that the three points of the triangle are \[A( - 5,6)\], \[B( - 4, - 2)\] and \[C(7,5)\] . We will find distance AB , BC and CA.
So, firstly we will calculate distance AB as \[A( - 5,6)\] and \[B( - 4, - 2)\] using distance formula which is
AB = \[\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]
 Here, \[{x_1} = - 5\] , \[{x_2} = - 4\] , \[{y_1} = 6\] and \[{y_2} = - 2\] .
On substituting the values we get,
Then, AB = \[\sqrt {{{\left( { - 4 + 5} \right)}^2} + {{\left( { - 2 - 6} \right)}^2}} \]
On simplifying:
Then, AB = \[\sqrt {{{\left( 1 \right)}^2} + {{\left( { - 8} \right)}^2}} \]
By opening the squares and adding we get;
Then, AB = \[\sqrt {1 + 64} \]
Hence, AB = \[\sqrt {65} \]
So, secondly we will calculate distance BC as \[B( - 4, - 2)\] and \[C(7,5)\] using distance formula which is
BC = \[\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]
 Here, \[{x_1} = - 4\] , \[{x_2} = 7\] , \[{y_1} = - 2\] and \[{y_2} = 5\] .
On substituting the values we get,
Then, BC = \[\sqrt {{{\left( {7 + 4} \right)}^2} + {{\left( {5 + 2} \right)}^2}} \]
On simplifying:
Then, BC = \[\sqrt {{{\left( {11} \right)}^2} + {{\left( 7 \right)}^2}} \]
By opening the squares and adding we get;
Then, BC = \[\sqrt {121 + 49} \]
Hence, BC = \[\sqrt {170} \]\[\]
So, firstly we will calculate distance AB as \[C(7,5)\] and \[A( - 5,6)\] using distance formula which is
CA = \[\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]
 Here, \[{x_1} = 7\] , \[{x_2} = - 5\] , \[{y_1} = 5\] and \[{y_2} = 6\] .
On substituting the values we get,
Then, BC = \[\sqrt {{{\left( { - 5 - 7} \right)}^2} + {{\left( {6 - 5} \right)}^2}} \]
On simplifying:
Then, BC = \[\sqrt {{{\left( {12} \right)}^2} + {{\left( 1 \right)}^2}} \]
By opening the squares and adding we get;
Then, BC = \[\sqrt {144 + 1} \]
Hence, BC = \[\sqrt {145} \]
As, it is clear from the above \[AB \ne BC \ne CA\] .

Hence, all the sides are different. So, the triangle formed is scalene. As, the length of the sides are not equal.

Note:
To solve these types of questions, you need to remember the distance formula. As well as you need to know the difference between the scalene, equilateral and isosceles triangle respectively which is formed after calculating the distance.