Question

In the following example, can the segment joining the given point form a triangle? If a triangle is formed, state the type of the triangle considering the side of the triangle.

L(6, 4), M(–5, –3), N(–6, 8)

Answer 1

L(6, 4), M(–5, –3), N(–6, 8)

By distance Formula,

d(L, M) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

= `sqrt((-5-6)^2 + (-3-4)^2)`

= `sqrt((-11)^2 + (-7)^2)`

= `sqrt(121 + 49)`

= `sqrt(170)`

∴ d(L, M) = `sqrt(170)`          ......(i)

By distance Formula,

d(M, N) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

= `sqrt([-6 - (-5)]^2 + [8 - (- 3)]^2)`

= `sqrt((-6 + 5)^2 + (8 + 3)^2)`

= `sqrt((-1)^2 + (11)^2)`

= `sqrt(1 + 121)`

= `sqrt(122)`

∴ d(M, N) =  `sqrt(122)`            ......(ii)

By distance Formula,

d(L, N) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

= `sqrt((-6-6)^2 + (8 - 4)^2)`

= `sqrt((-12)^2 + (4)^2)`

= `sqrt(144 + 16)`

= `sqrt(160)`

∴ d(L, N) = `sqrt(160)`        ........(iii)

On adding (ii) and (iii)

∴ d(M, N) + d (L, N) > d (L, M)

∴ Points L, M, N are non collinear points.
∴ We can construct a triangle through 3 non-collinear points.
Since LM ≠ MN ≠ LN

∴ ΔLMN is a scalene triangle.

∴ The segments joining the points L, M and N will form a scalene triangle.