Hushar Mulga
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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. (1) L(6,4) , M(-5,-3) , N(-6,8)

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.

(1) L(6,4) , M(-5,-3) , N(-6,8)

L(6, 4), M(–5, –3), N(–6, 8)

By distance Formula,

By distance Formula,

LM = \[\sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}\]

= \[\sqrt{((-5-6)^2 + (-3-4)^2)}\]

= \[\sqrt{((-11)^2 + (-7)^2)}\]

= \[\sqrt{(121 + 49)}\]

= \[\sqrt{(170)}\]

∴ LM = \[\sqrt{(170)}      ......(i)\]

By distance Formula,

MN = \[\sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}\]

= \[\sqrt{((-6 - (-5))^2 + (8 - (- 3))^2)}\]

= \[\sqrt{((-1)^2 + (11)^2)}\]

= \[\sqrt{(1 + 121)}\]

= \[\sqrt{(122)}\]

∴ MN =  \[\sqrt{(122)}        ......(ii)\]

By distance Formula,

LN = \[\sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}\]

= \[\sqrt{((-6-6)^2 + (8 - 4)^2)}\]

= \[\sqrt{((-12)^2 + (4)^2)}\]

= \[\sqrt{(144 + 16)}\]

= \[\sqrt{(160)}\]

\[∴ LN = \sqrt{(160)}    ........(iii)\]

(MN + LN) > LM

These points are not collinear.
∴ We can construct a triangle through 3 non-collinear points.
LM ≠ MN ≠ LN

∴ Triangle formed is a scalene triangle.

Explanation:-

Given points are L(6, 4), M(-5, -3), and N(-6, 8).

Using the distance formula, we can find the length of each side of the triangle.

The distance formula is given by:

d = √[(x2 – x1)^2 + (y2 – y1)^2]

Where d is the distance between two points (x1, y1) and (x2, y2).

Finding LM:

LM = √[(-5 – 6)^2 + (-3 – 4)^2]

= √[(-11)^2 + (-7)^2]

= √[121 + 49]

= √170

Finding MN:

MN = √[(-6 – (-5))^2 + (8 – (-3))^2]

= √[(-1)^2 + (11)^2]

= √[1 + 121]

= √122

Finding LN:

LN = &radic[(-6 – 6)^2 + (8 – 4)^2]

= √[(-12)^2 + (4)^2]

= √[144 + 16]

= √160

As (MN + LN) > LM, the given points are not collinear, and hence, we can construct a triangle through these points.

Also, since LM ≠ MN ≠ LN, the triangle formed is a scalene triangle.

Chapter 5. Co-ordinate Geometry – Problem set 5 (Page 122)