Find the terminal point.

Question

The line segment from (1, 4) to (2, 1) is extended a distance equal to twice its length.

Answer 1

Look at the slope between the original points (1,4) and (2,1)

m = (y₂-y₁)/(x₂-x₁) = (1-4)/(2-1) = -3/1

This means that to find a new point on the line we go right

one space for x and down 3 spaces for y

(x₂,y₂) = (x₁+1, y₁-3)

 

If we extend the line a distance equal to twice it's length

the new line is 3 times as long as the original line

 

We need to find the next two points on the line...

Starting from the original terminus point of (2,1)

(x₂,y₂) = (x₁+1, y₁-3) = (2+1,1-3) = (3,-2)

(x₃,y₃) = (x₂+1, y₂-3) = (3+1,-2-3) = (4,-5)

 

The point (4,-5) is the new terminal point.

 

We can check by verifying we have tripled the length of

the original line.

distance between two points is  d = √[(x₂-x₁)²+(y₂-y₁)²]

The distance from (1,4) to (2,1) = √[(2-1)²+(1-4)²] = √10

 

If our new end point is (4,-5) the distance from

(1,4) to (4,-5) should be 3√10 or √90

 

d = √[(4-1)²+(-5-4)²] = √[(3)²+(-9)²] = √(9+81) = √90

 

Our distance is correct.