Verifying a Point Lies on a Line
Verifying a point lies on a line is a Grade 8 math skill covered in Chapter 2: Dilations, Similarity, and Introducing Slope. A point (x3, y3) lies on the line through (x1, y1) and (x2, y2) if the slope from (x1, y1) to (x3, y3) equals the slope of the line. Students compute the slope ratio (change in y) / (change in x) for both segments and compare.
Key Concepts
Property A point $(x 3, y 3)$ lies on the line passing through points $(x 1, y 1)$ and $(x 2, y 2)$ if the slope between $(x 1, y 1)$ and $(x 3, y 3)$ is the same as the slope of the line. $$m = \frac{y 2 y 1}{x 2 x 1} = \frac{y 3 y 1}{x 3 x 1}$$.
Examples Does the point $(6, 7)$ lie on the line that passes through $(0, 1)$ and $(3, 4)$? The slope of the line is $m = \frac{4 1}{3 0} = \frac{3}{3} = 1$. The slope from $(0, 1)$ to $(6, 7)$ is $m = \frac{7 1}{6 0} = \frac{6}{6} = 1$. Since the slopes are equal, the point $(6, 7)$ is on the line. Does the point $(2, 5)$ lie on the line that passes through $( 1, 1)$ and $(1, 3)$? The slope of the line is $m = \frac{3 ( 1)}{1 ( 1)} = \frac{4}{2} = 2$. The slope from $( 1, 1)$ to $(2, 5)$ is $m = \frac{5 ( 1)}{2 ( 1)} = \frac{6}{3} = 2$. Since the slopes are equal, the point $(2, 5)$ is on the line.
Explanation This skill uses the core concept that the slope is constant everywhere on a line. To check if a specific point is on a line, you can calculate the slope of the original line using two given points. Then, calculate the slope between one of the given points and the point you are testing. If the slopes are identical, the test point must lie on the same line.
Common Questions
How do you verify that a point lies on a line?
Calculate the slope from one known point on the line to the new point. If it equals the slope of the line, the new point lies on the line.
What formula checks if a point is on a line?
Check that (y3 - y1) / (x3 - x1) equals the known slope m = (y2 - y1) / (x2 - x1). If the ratios are equal, the point is on the line.
Does (4, 7) lie on the line through (0, 1) and (2, 4)?
Slope from (0,1) to (2,4): (4-1)/(2-0) = 3/2. Slope from (0,1) to (4,7): (7-1)/(4-0) = 6/4 = 3/2. Equal, so yes, (4,7) is on the line.
Where is verifying a point lies on a line taught in Grade 8?
Chapter 2: Dilations, Similarity, and Introducing Slope in 8th grade math.
What does it mean if the slopes are not equal when checking a point?
If the slope from the line to the new point differs from the line slope, the point is not on the line; it is somewhere off the line.