Question

9. Show that the following sets of
points are collinear
) (2,5), (4,6) and (8,8)​

Answer 1

hope it's helpful for you

Answer 2

Answer:

There are many methods to prove that a given set of points are collinear. I have given the two prominently used methods along with the concept. Hope it helps you!!

Method 1:  Area of triangle = 0

In this method, the 3 set of points are assumed to be the coordinates of a triangle. Finding the area of the triangle formed, if they turn out to be zero, then the given set of points is collinear. If the Area is greater than zero, then the points are not collinear.

Method 2:  AB + BC = AC

If the given set of Points are considered to be A, B and C, then they are collinear if the condition:

→ AB + BC = AC is true.

Else, they aren't collinear.

Let us solve this question using Method 2:

Given points are:

• A ( 2, 5 ) ; B ( 4, 6 ) ; C ( 8, 8 )

Calculating the distance we get:

$\implies AB = \sqrt{ ( 4 - 2 )^2 + ( 6 - 5 )^2 }\\\\\\\implies AB = \sqrt{ 2^2 + 1^2} = \sqrt{ 4 + 1 } \\\\\\\implies \boxed{ AB = \sqrt{5}}\\\\\\\implies BC = \sqrt{ (8-4)^2 + (8-6)^2}\\\\\\\implies BC = \sqrt{ 4^2 + 2^2 } = \sqrt{ 16 + 4 }\\\\\\\implies \boxed{ BC = \sqrt{20} = 2 \sqrt{5} }\\\\\\\implies AC = \sqrt{ (8-2)^2 + (8-5)^2}\\\\\\\implies AC = \sqrt{ 6^2 + 3^2 }\\\\\\\implies \boxed{AC = \sqrt{45} = 3 \sqrt{5}}$

Substituting the values in AB, BC and AC we get:

⇒ √5 + 2√5 = 3√5

⇒ 3√5 = 3√5

⇒ LHS = RHS

Hence Proved!