Question

find the value of k for which. the points r collinear
i) (2,5), (4, k) and (8,8)

I want with solution​

Answer 1

Answer:

Let the given points A=(2,5) B=(4,K) C=(8,8)

We can say that area of ABC=0

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

Here x1=2,x2=4,x3=8

y1=5, y2=K,y3=8

Putting values

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

1/2[2k-16+12+40-8k]=0

1/2[36-6k]=0

18-3k=0

k=-18/-3

k=6

Answer 2

Three colinear points need the same slope.

We get two slopes from the given points.

$\boxed{\sf{Slope\;of\;(2,\;5)\;and\;(8,\;8)}}$

$\sf{\implies \dfrac{\Delta y}{\Delta x} =\dfrac{8-5}{8-2} =\dfrac{3}{6} }$

$\sf{\therefore \dfrac{\Delta y}{\Delta x} =\dfrac{1}{2} }$

$\boxed{\sf{Slope\;of\;(4,\;k)\;and\;(8,\;8)}}$

$\sf{\implies \dfrac{\Delta y}{\Delta x} =\dfrac{8-k}{8-4} }$

$\sf{\therefore \dfrac{\Delta y}{\Delta x} =\dfrac{8-k}{4} }$

Hence we get an equation for one variable.

Two slopes need to be equal.

$\sf{\implies \dfrac{8-k}{4} =\dfrac{1}{2} }$

$\sf{\implies 8-k=4}$

$\sf{\therefore k=4}$

Side note: I've attached the reasoning of the slope approach. I approached with the slope, but you can find the solution based on the triangle area formula. Hope this helps you.