# In each of the following find the value of ‘k’, for which the points are collinear.

(i) (7, - 2), (5, 1), (3, k) (ii) (8, 1), (k, - 4), (2, - 5)

**Solution:**

Three or more points are said to be collinear if they lie on a straight line.

(i) Let A(x₁, y₁) = (7, - 2), B(x₂, y₂) = (5 , 1) and C(x₃, y₃) = (3, k)

For three points to be collinear, the area of triangle must be equal to zero.

1/2 [x₁ (y₂ - y₃) + x₂ (y₃_{ }- y₁) + x₃ (y₁ - y₂)] = 0

By substituting the values of vertices, A, B, C in the above formula,

1/2 [7{1 - k} + 5{k - (- 2)} + 3{(- 2) - 1}] = 0

7 - 7k + 5k + 10 - 9 = 0

- 2k + 8 = 0

k = 4

Hence, the given points are collinear for k = 4

(ii) Let A(x₁, y₁) = (8, 1), B(x₂, y₂) = (k , - 4) and C(x₃, y₃) = (2, - 5)

For three points to be collinear, the area of triangle must be equal to zero.

1/2 [x₁ (y₂ - y₃) + x₂ (y₃_{ }- y₁) + x₃ (y₁ - y₂)] = 0

By substituting the values of vertices, A, B, C in the above formula,

1/2 [8{- 4 - (- 5)} + k{(- 5) - (1)} + 2{1 - ( - 4)}] = 0

8 - 6k + 10 = 0 (By Transposing)

6k = 18

k = 3

Hence, the given points are collinear for k = 3

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 7

**Video Solution:**

## In each of the following find the value of ‘k’, for which the points are collinear. (i) (7, - 2), (5, 1), (3, k) (ii) (8, 1), (k, - 4), (2, - 5)

NCERT Class 10 Maths Solutions Chapter 7 Exercise 7.3 Question 2

**Summary:**

In each of the following the value of ‘k’, for which the points are collinear are: (i) (7, - 2), (5, 1), (3, k) ; k = 4 (ii) (8, 1), (k, - 4), (2, - 5) ; k = 3

**☛ Related Questions:**

- Find the area of the triangle whose vertices are:(i) (2, 3), (- 1, 0), (2, - 4)(ii) (- 5, - 1), (3, - 5), (5, 2)
- Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, - 1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.
- Find the area of the quadrilateral whose vertices, taken in order, are (- 4, - 2), (- 3, - 5), (3, - 2) and (2, 3).
- You have studied in Class IX that a median of a triangle divides it into two triangles of equal areas. Verify this result for ∆ABC whose vertices are A (4, - 6), B (3, - 2) and C (5, 2)

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