Question

show by Section formula that the points ( 3, - 2 ) ,( 5, 2) and (8,8) are collinear

Answer 1

HELLO THERE!

Required to Prove: The points (3,-2), (5,2) and (8,8) are collinear, i.e., they lie on a same line.

Let:

A = (3,-2)

B = (5,2)

C = (8,8)

Let the point B divide AC in the ratio of k+1

Hence, the coordinates will be,

(8k+2/k+1, 8k+5/k+1)

Further, we know that coordinates of B are (4,6)

Comparing we get,

8k+2/k+1 = 4

and 8k+5/k+1 = 6

=> 8k+2 = 4k+4 and 8k+5 = 6k+6

=> 4k=2 and 2k=1

k=2/4 and k=1/2

k=1/2 and k=1/2

We get, that value of k is same in both x and y directions.Therefore Points A,B,C are collinears. (PROVED).

Other method to prove collinearity:

Find the slope of AB and BC (using the two points). If Slope of AB = Slope of BC, then A, B, C are collinear.

There is also a formula: x₁(y₂ - y₃) + x₂(y₃-y₁) + x₃(y₁ - y₂).

Put the values of (x₁, y₁), (x₂, y₂) and (x₃, y₃) in the equation. If the result is zero, then the points are collinear.

HOPE MY ANSWER IS SATISFACTORY..

THANKS!

Answer 2

Now we have to prove that the points (3,-2),(5,2) and(8,8) are collinear. Let B divides AC in the ratio k:1 Then the co ordinates of B is ((8k+3/k+1),(8k-2/k+1)) But the co ordinates of B is (5,2) Comparing we get ((8k+3/k+1)=5,(8k-2/k+1))=2 8k+3=5k+5 and 8k-2=2k+2 8k-5k=5-3 and 8k-2k=2+2 3k=2 and 6k=4 k=2/3 and k=4/6 =2/3 The value of k is same in both x and y coordinates. So the points A,B and C are collinear