8.EE.6: Slope and Similar Triangles
Also written as 8.EE.B.6 and CCSS.MATH.CONTENT.8.EE.B.6 · Cluster: Understand the connections between proportional relationships, lines, and linear equations.
The Official Wording
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Common Core State Standards, Grade 8, Expressions & Equations · Read it at the source · Checked October 14, 2026
What 8.EE.6 Means in Plain English
Eighth graders draw two slope triangles on a line and argue they are similar, so rise over run matches: 3/2 from (0, 1) to (2, 4), and 6/4 from (2, 4) to (6, 10). A triangle to any point (x, y) then gives y = (3/2)x + 1.
Through the origin the same argument gives y = mx, the proportional lines of 8.EE.5. This derivation sits under every y = mx + b in 8.F.3 and 8.F.4, and in high school slope is used to prove when lines are parallel or perpendicular (HSG-GPE.5).
8.EE.6 I Can Statements
- I can draw slope triangles and find rise over run.
- I can explain why similar triangles give the same slope anywhere.
- I can derive y = mx + b from a line’s slope and intercept.
8.EE.6 Worked Example
Two slope triangles on the same line, then a third to a general point that turns into the equation.
A line passes through (0, 1), (2, 4) and (6, 10). Use slope triangles to show its slope is the same between any two points, then derive its equation.
- Draw a slope triangle from (0, 1) to (2, 4): 2 across, then 3 up.rise 3, run 2, slope 3/2
- Draw a bigger one from (2, 4) to (6, 10): 4 across, then 6 up.rise 6, run 4, slope 6/4 = 3/2
- Both triangles have a right angle, and their angles where the line meets the horizontal are equal because the horizontals are parallel. Two equal angles make them similar, so their sides are in proportion.6/4 = 3/2: the same slope between any two points
- Now draw a triangle from (0, 1) to any point (x, y) on the line. Its run is x and its rise is y - 1, and its slope must also be 3/2.(y - 1)/x = 3/2
- Multiply both sides by x, then add 1 to both sides.y = (3/2)x + 1
Check it: Substitute the given points: (3/2)(2) + 1 = 4 and (3/2)(6) + 1 = 10, so both (2, 4) and (6, 10) are on the line.
Common Mistakes with 8.EE.6
Dividing coordinates instead of differences
Slope from (2, 4) to (6, 10) written as 10/6
Try this: Subtract before dividing: the rise is 10 - 4 = 6, the run is 6 - 2 = 4, and 6/4 = 3/2.
Thinking a bigger triangle means a steeper line
“The triangle from (2, 4) to (6, 10) is bigger, so the line is steeper there.”
Try this: Compare the ratios, not the sizes: 3/2 and 6/4 are equal, because similar triangles stretch rise and run by the same factor.
Mixing up m and b
A line with slope 3/2 crossing the vertical axis at 1 written y = x + 3/2
Try this: Find b where the line crosses the vertical axis, at x = 0; m is the rise for each one step right.
Where 8.EE.6 Fits
Practice 8.EE.6
Quick Check
- A line passes through (1, 5) and (4, 11). What is its slope?
- A line through the origin has slope 4/5. Write its equation.
- A line has slope -2 and crosses the vertical axis at 7. Does it pass through (3, 1)?