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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)
Solution
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.
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