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 )
Solution
[Aleft( sqrt{2} , sqrt{2} right), Bleft( -sqrt{2} , -sqrt{2} right), Cleft( -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)