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.
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
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`
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
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
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.