Use the distance formula to determine whether the points lie on the same line.
(0,4),(7,-6),(-5,11).
I know the distance formula, but I don't see how that will help me find out if they lie on the same line.
distance between (0,4) and (7,-6) = √(49+100) = √149
distance between (7,-6) and (-5,11) = √(144+289) = √433
distance between (0,4) and (-5,11) = √(25+49) = √74
if they are collinear, then
√74 + √149 = √433
IS it ??
√74 + √149 = 20.80888088
√433 = 20.80865205
difference is .000228
had you simply graphed it, you would not have noticed the difference, but there is a difference, so the points are NOT collinear.
check: slope of 1st segment = -10/7
slope of another segment = 7/-5
since -10/7 ≠ -7/5 , they are not collinear.
(notice the slopes are close)
-1.42857 vs -1.4 , angles are 55.0° vs 54.5°
Well, if you are trying to determine if three points lie on the same line, you can use the distance formula to check if the distances between any two points are proportional. However, since you mentioned you don't see how that will help, let me try a different approach.
To determine if the points (0,4), (7,-6), and (-5,11) lie on the same line, we can calculate the slopes between any two pairs of points and see if they are equal.
The slope between (0,4) and (7,-6) is given by (y2 - y1) / (x2 - x1).
So, (−6 - 4) / (7 - 0) = -10 / 7.
The slope between (0,4) and (-5,11) is given by (y2 - y1) / (x2 - x1).
So, (11 - 4) / (-5 - 0) = 7 / -5 = -7 / 5.
The slope between (7,-6) and (-5,11) is given by (y2 - y1) / (x2 - x1).
So, (11 + 6) / (-5 - 7) = 17 / -12 = -17 / 12.
Since the slopes between the three pairs of points are not equal, the points (0,4), (7,-6), and (-5,11) do not lie on the same line. They may form a sort of triangle, or maybe a squiggly line. Who knows? Geometry can be quite unpredictable!
To determine whether the points (0,4), (7,-6), and (-5,11) lie on the same line, you can use the distance formula to calculate the distances between each pair of points and see if they satisfy the condition for collinearity.
The distance formula is given by:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Let's first calculate the distance between (0,4) and (7,-6):
Distance = √[(7 - 0)^2 + (-6 - 4)^2]
= √[49 + 100]
= √149
Next, calculate the distance between (0,4) and (-5,11):
Distance = √[(-5 - 0)^2 + (11 - 4)^2]
= √[25 + 49]
= √74
Finally, calculate the distance between (7,-6) and (-5,11):
Distance = √[(-5 - 7)^2 + (11 - (-6))^2]
= √[(-12)^2 + (17)^2]
= √[144 + 289]
= √433
Now, compare the distances:
√149 ≠ √74 ≠ √433
Since the distances are not equal, the points (0,4), (7,-6), and (-5,11) do not lie on the same line.
To determine whether the points (0,4), (7,-6), and (-5,11) lie on the same line using the distance formula, we can calculate the distances between the three pairs of points.
Let's label the points as follows:
Point A = (0,4)
Point B = (7,-6)
Point C = (-5,11)
Distance between A and B:
d(AB) = √((x₂ - x₁)² + (y₂ - y₁)²)
= √((7 - 0)² + (-6 - 4)²)
= √(7² + (-10)²)
= √(49 + 100)
= √149
≈ 12.21
Distance between A and C:
d(AC) = √((x₂ - x₁)² + (y₂ - y₁)²)
= √((-5 - 0)² + (11 - 4)²)
= √((-5)² + 7²)
= √(25 + 49)
= √74
≈ 8.60
Distance between B and C:
d(BC) = √((x₂ - x₁)² + (y₂ - y₁)²)
= √((-5 - 7)² + (11 - (-6))²)
= √((-12)² + 17²)
= √(144 + 289)
= √433
≈ 20.82
Now, if the points A, B, and C lie on the same line, the sum of any two distances would be equal to the third distance. However, if the sum of any two distances is not equal to the third distance, then the points do not lie on the same line.
Let's check if the sum of any two distances equals the third distance:
d(AB) + d(AC) ≈ 12.21 + 8.60 ≈ 20.81
d(AB) + d(BC) ≈ 12.21 + 20.82 ≈ 33.03
d(AC) + d(BC) ≈ 8.60 + 20.82 ≈ 29.42
Since none of the sums of distances equal each other, we can conclude that the points (0,4), (7,-6), and (-5,11) do not lie on the same line.