Using the distance formula I have to prove that ABC are colinear.
A(-1, -2) B(2, 2) C(8,10)
I got the square root of 17 for line AB
And I got the square root of 244 for line BC, but already this can't make sense?
To prove that points A, B, and C are collinear using the distance formula, we need to show that the slopes of the lines AB and BC are equal. If the slopes are equal, it indicates that the points are located on the same straight line.
Let's begin by finding the slopes of the lines AB and BC using the coordinates given:
Line AB:
Let (x₁, y₁) be the coordinates of point A (-1, -2), and (x₂, y₂) be the coordinates of point B (2, 2).
The slope of line AB can be calculated using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Substituting the values, we have:
m₁ = (2 - (-2)) / (2 - (-1))
= 4 / 3
= 1.3333...
Line BC:
Let (x₃, y₃) be the coordinates of point B (2, 2), and (x₄, y₄) be the coordinates of point C (8, 10).
The slope of line BC can be calculated using the formula:
m = (y₄ - y₃) / (x₄ - x₃)
Substituting the values, we have:
m₂ = (10 - 2) / (8 - 2)
= 8 / 6
= 1.3333...
As we can see, the slopes of line AB and line BC are both equal to 1.3333..., which indicates that points A, B, and C lie on the same straight line. Therefore, ABC is colinear.
Now, let's address your question regarding the distances. The distance formula is used to calculate the distance between two points in a plane. It is not directly related to proving collinearity.
To find the distance between two points P(x₁, y₁) and Q(x₂, y₂), we can use the distance formula:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
Let's calculate the distances AB and BC using the given coordinates:
Distance AB:
d₁ = √((2 - (-1))² + (2 - (-2))²)
= √(3² + 4²)
= √(9 + 16)
= √25
= 5
Distance BC:
d₂ = √((8 - 2)² + (10 - 2)²)
= √(6² + 8²)
= √(36 + 64)
= √100
= 10
We can see that the distance AB is indeed 5 units, and the distance BC is indeed 10 units. However, the distances themselves do not prove or disprove collinearity. The slopes of the lines AB and BC should be examined to determine collinearity.
In conclusion, the points A, B, and C are collinear because the slopes of lines AB and BC are equal, regardless of the specific distances involved.