# WORKSHEET ON COLLINEAR POINTS

## About "Worksheet on collinear points"

Worksheet on collinear points is much useful to the students who would like to practice problems on collinearity.

## Worksheet on collinear points - Problems

1)  Using the concept of area of triangle, show that the            points A(5, -2), B(4, -1) and C(1, 2) are collinear.

2)  Using the concept of distance between two points,             show that the points A(5, -2), B(4, -1) and C(1, 2) are              collinear.

3)  Using the concept of equation of line, show that the             points A(5, -2), B(4, -1) and C(1, 2) are collinear.

3)  Using the concept of slope, show that the points                  A(5, -2), B(4, -1) and C(1, 2) are collinear. ## Worksheet on collinear points - Answers

Problem 1 :

Using the concept of area of triangle, show that the points A(5, -2), B(4, -1) and C(1, 2) are collinear.

Solution :

Using the concept of area of triangle, if the three points A(x₁, y₁), B(x₂, y₂) and C (x₃, y₃)  are collinear, then

x₁y₂ + x₂y₃ + x₃y   =   xy + x₃y₂ + xy₃

Comparing the given points to A(x₁, y₁), B(x₂, y₂) and C (x₃, y₃), we get

(x₁, y₁)  =  (5, -2)

(x₂, y₂)  =  (4, -1)

(x₃, y₃)  =  (1, 2)

x₁y₂ + x₂y₃ + x₃y₁  =  5x(-1) + 4x2 + 1x(-2)

x₁y₂ + x₂y₃ + x₃y₁  =  -5 + 8 -2

x₁y₂ + x₂y₃ + x₃y₁  =  1 --------(1)

xy + x₃y₂ + xy₃  =  4x(-2) + 1x(-1) + 5x(2)

xy + x₃y₂ + xy₃  =  -8 - 1 + 10

xy + x₃y₂ + xy₃  =  1 --------(2)

From (1) and (2), we get

x₁y₂ + x₂y₃ + x₃y   =   xy + x₃y₂ + xy₃

Hence, the three points A, B and C are collinear.

Let us look at the next problem on "Condition of collinearity of three points"

Problem 2 :

Using the concept of distance between two points, show that the points A(5, -2), B(4, -1) and C(1, 2) are collinear.

Solution :

We know the distance between the two points (x₁, y₁) and (x₂, y₂) is

d  =  √ (x₂ - x₁) ² + (y₂ - y₁) ²

Let us find the lengths AB, BC and AC using the above distance formula.

AB  =  √ [(4 - 5)² + (-1 + 2)²]

AB  =  √ [(-1)²  + (1)²]

AB  =  √ [1 + 1]

AB  =  √2

BC  =  √ [(1 - 4)² + (2 + 1)²]

BC  =  √ [(-3)²  + (3)²]

BC  =  √ [9 + 9]

BC  =  √[2X9]

BC  =  3√2

AC  =  √ [(1 - 5)² + (2 + 2)²]

AC  =  √ [(-4)²  + (4)²]

AC  =  √ [16 + 16]

AC  =  √[2x16]

AC  =  4√2

Therefore, AB + BC  =  √2 + 3√2  =  4√2  =  AC

Thus, AB + BC  =  AC

Hence, the given three points A, B and C are collinear.

Let us look at the next problem on "Condition of collinearity of three points"

Problem 3 :

Using the concept of equation of line, show that the points A(5, -2), B(4, -1) and C(1, 2) are collinear.

Solution :

Equation of the straight line in two-points form is

(y - y₁) / (y₂ - y₁)  =  (x - x₁) / (x₂ - x₁)

Using the above formula, let us get equation of the line through the points A and B.

Plugging (x₁ , y₁) = (5, -2) and (x₂, y) = (4, -1), we get

(y +2) / (-1 + 2)  =  (x - 5) / (4 - 5)

(y + 2) / 1  =  (x - 5) / (-1) ----------> y +2  =  -x + 5

y +2  =  -x + 5 ----------> x + y - 3  =  0

Now, we can plug the third point C(1, 2) in the above equation.

That is, plug x = 1 and y = 2

x + y - 3 = 0 ---------->  1 + 2 - 3  =  0 -------> 0 = 0

Therefore, the third point C(1, 2) satisfies the equation.

Hence, the given points three points A, B and C are collinear.

Problem 4 :

Using the concept of slope, show that the points A(5, -2), B(4, -1) and C(1, 2) are collinear.

Solution :

Slope of the line joining (x₁, y₁) and (x₂, y₂) is,

m  =  (y₂ - y₁) / (x₂ - x₁)

Using the above formula,

Slope of the line AB joining the points A (5, - 2) and B (4- 1) is

=  (-1 + 2) / (4 - 5)

=  - 1

Slope of the line BC joining the points B (4- 1) and C (1, 2) is

=  (2 + 1) / (1 - 4)

=  - 1

Thus, slope of AB = slope of BC.

And also, B is the common point.

Hence, the points A , B and C are collinear.

Let us look at the next problem on "Condition of collinearity of three points"

After having gone through the stuff given above, we hope that the students would have understood "Worksheet on collinear points".

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