Please answer this question for I have a test coming up. My question is: Q. Given that the points (1,-1), (2,2) and (4,t) are collinear, find the value of t. This is Add maths question and hope...

Please answer this question for I have a test coming up.

My question is:

Q. Given that the points (1,-1), (2,2) and (4,t) are collinear, find the value of t.

This is Add maths question and hope to see your answer most probably today.

Expert Answers
uwais17 eNotes educator| Certified Educator

heFirstly it is important to know what collinear means. Collinear is when 3 or more points lie on the same straight line. 

When approaching this problem it is essential to use what is given and what is required. 

We are given:

  • Two points with both x and y values
  • A third point only with one x value 

We are required: 

  • Determine t, the y value if third point

Approach:

1. Determine the equation of the straight line: 

In order to determine the equation of the straight line we need two points. Hence we will use the points: `(1,-1), (2,2)'`

First determine the gradient of the equation, gradient is denoted as 'm':

`m=(y2-y1)/(x2-x1) =(2+1)/(2-1)= 3`

use the following to determine the equation of the line:

`y - y1 = m (x - x1)`

You can substitute one point and the gradient:

`y - 2 = 3 (x -2)`

`y -2 = 3x - 6`

`y = 3x -4`

2. Now use the equation of the line to determine t by subtituting x =4:

`t= 3(4) -4 = 12 -4 = 8`

Seema Adiga eNotes educator| Certified Educator

If three points are collinear the area of the triangle formed by them is 0.

The formula to find area of a triangle formed by three points is

 

`1/2 [x_1 (y_2 - y_3) + x_2 (y_3 - y_1) + x_3(y_1 - y_2)] `

Now the given points `(1,-1), (2,2)` and `(4,t)` are colliner. So the area is 0

Lets take these points as `(x_1, y_1), (x_2, y_2), (x_3, y_3)` and substitute in the formula

 

`1/2 [1(2-t) + 2(t+1) + 4(-1 -2)] =0 `

 

`2 - t + 2t + 2 - 12 = 0 `

 

`t - 8 = 0 `

 

`t = 8 `

The required answer is t = 8

Borys Shumyatskiy eNotes educator| Certified Educator

Hello!

Let's write an equation of the straight line that goes through the given points (1, -1) and (2, 2). Then the third point must satisfy this equation.

For two points (`x_1,` `y_1`) and (`x_2,` `y_2`) the equation is

`(x-x_1)/(x_1-x_2)=(y-y_1)/(y_1-y_2).`

 

In our case it is

`(x-1)/(1-2)=(y-(-1))/(-1-2),` or `x-1=(y+1)/3.`

 

The third point has `x=4` and `y=t,` so

`4-1=(t+1)/3,` or `3*3=t+1,` so t=8. This is the answer.