In the following examples, can the segment joining the given points form a triangle ?
8. In the following examples, can the segment joining the given points form a triangle ? If triangle is formed, state the type of the triangle considering sides of the triangle.
(3) A( 2 , 2 ), B( – 2 , – 2 ), C( – 6 , 6 )
\[A\left( \sqrt{2} , \sqrt{2} \right), B\left( -\sqrt{2} , -\sqrt{2} \right), C\left( -\sqrt{6} , \sqrt{6} \right)\]
\[AB = \sqrt{\left( \sqrt{2} + \sqrt{2} \right)^2 + \left( \sqrt{2} + \sqrt{2} \right)^2} = \sqrt{8 + 8} = \sqrt{16} = 4\]
\[BC = \sqrt{\left( - \sqrt{2} + \sqrt{6} \right)^2 + \left( - \sqrt{2} - \sqrt{6} \right)^2} = \sqrt{16} = 4\]
\[AC = \sqrt{\left( \sqrt{2} + \sqrt{6} \right)^2 + \left( \sqrt{2} - \sqrt{6} \right)^2} = \sqrt{16} = 4\]
\[AB = BC = AC\]
So, these vertices form an equilateral triangle.
Explanation:-
The given points are A(√2, √2), B(-√2, -√2), and C(-√6, √6).
To find the length of AB, we can use the distance formula: AB = √[(√2 – (-√2))^2 + (√2 – (-√2))^2] = √[4 + 4] = √8 * √2 = 2√2 * √2 = 4
Similarly, we can find the length of BC and AC using the distance formula: BC = √[(-√2 – (-√6))^2 + (-√2 – √6)^2] = √[4 + 4] = 4
AC = √[(√2 – (-√6))^2 + (√2 – √6)^2] = √[16] = 4
Since all three sides have the same length, the triangle is equilateral. Therefore, the given points form an equilateral triangle.
Chapter 5. Co-ordinate Geometry – Problem set 5 (Page 122)