Q) Using distance formula, prove that the points A (2, 3), B (- 7, 0) and C (- 1,2) are collinear.

(Q 25 B – 30/5/2 – CBSE 2026 Question Paper)

Ans:

[Approach: To prove that three points are collinear using the distance formula, we must show that the distance between first two points plus the distance between 2nd & 3rd points equals the total distance between the two outermost points (1st & 3rd). i.e. If points A, B and C are collinear, then AB + BC = AC, if B lies between A & C]

Step 1: Here, We have the points A (2, 3) B (- 7, 0) and C (- 1,2)

We can see that C lies between A and B and now we will check collinearity of A, C and B.

and hence we will check if AC + CB = AB or not.

Step 2: Let’s start by calculating distances AC, CB and AB.

We know that according to the distance formula, distance between 2 points (x1,y1) and (x2,y2)

D =

Step 3: ∴ Distance between A (2, 3) and C (- 1,2)

AC =

=

= √(9 + 1) = √10 units

Step 4: Similarly, Distance between C (- 1, 2) and B (- 7, 0)

CB =

=  = √(36 + 4)

= √40 = 2 √10 units

Step 5: and Distance between A (2, 3) B (- 7, 0)

AB =

=  = √(81 + 9)

= √90 = 3√10 units

Step 6: Now, Let’s check if AC + CB = AB

by LHS = AC + CB

= √10 + 2√10 = 3 √10

Since it matches with value of AB in RHS

Therefore, points A,B and C are collinear.

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