Opening subject page...
Loading your content
Discover how similar triangles prove that slope stays constant and lead to the equations of every line.
For thousands of years, people have wanted to describe straight-line patterns using math. Ancient builders needed to measure the steepness of ramps. Farmers tracked how crop yields changed over time. The idea of slope (how steep something is) has been around since ancient civilizations.
Over centuries, mathematicians found clever ways to turn the idea of steepness into exact equations. Let's look at some key moments in that journey.
Here is the big question this lesson answers: Why is the slope the same no matter which two points you pick on a line? We will use similar triangles to prove it, and then build the equations y = mx and y = mx + b step by step.
Before we dive in, you need to know four key ideas. Each one is a building block for the bigger proof.
The diagram below shows a line on the coordinate plane. We pick two different pairs of points and draw right triangles under the line. Notice how the triangles have the same shape but different sizes — they are similar triangles.
Here is the key idea: both triangles share the same angle where the line meets the horizontal leg. Both have a right angle (90°) where the vertical leg meets the horizontal leg. Since two angles match, the third angle must match too. That means the triangles are similar. And in similar triangles, the ratio of matching sides is always equal. So rise₁ ÷ run₁ = rise₂ ÷ run₂. That ratio is the slope.
Now let's turn the idea of constant slope into actual equations. We will start with the simplest case — a line through the origin (the point (0, 0)) — and then handle the general case.
Pick any point (x, y) on a line that passes through (0, 0). The slope between the origin and that point is:
Multiply both sides by x to solve for y:
What if the line does not pass through the origin? Suppose it crosses the y-axis at the point (0, b). Pick any other point (x, y) on the line. The slope between (0, b) and (x, y) is:
Multiply both sides by x: m × x = y − b. Then add b to both sides:
Let's walk through the full proof one more time using a clear diagram. The diagram below shows a line with y-intercept b and three points. We will form two slope triangles and show they give the same slope.
Step by step, here is what the diagram proves. First, the two right triangles both sit on the same line and share the same angle where the line meets the horizontal. Second, both have a 90° right angle. By the Angle-Angle (AA) rule, the triangles are similar. Third, because they are similar, the ratio rise ÷ run is the same in both. That ratio is the slope m.
Finally, using the y-intercept point (0, 1) and any point (x, y), we write m = (y − 1) ÷ x. Solving for y gives y = 1.5x + 1. This matches the form y = mx + b with m = 1.5 and b = 1.
A line passes through the points (0, −2) and (3, 4). Find the slope and write the equation of the line in slope-intercept form.
You now know two forms of a linear equation. Let's compare them side by side to make sure you understand when to use each one.
| Feature | y = mx | y = mx + b |
|---|---|---|
| Passes through the origin? | Yes — always goes through (0, 0) | Only if b = 0; otherwise it crosses at (0, b) |
| Y-intercept | b = 0 (hidden) | b can be any number |
| Number of parameters | 1 (just m) | 2 (m and b) |
| Example | y = 3x | y = 3x + 5 |
| Real-world use | Situations starting at zero, like earning $8/hour with no bonus | Situations with a starting value, like a $20 membership fee plus $8/hour |
The ideas you learned today are the starting point for many things you will study later. Here is a preview of how slope-intercept form connects to more advanced math.
| What You Know Now | What Comes Next |
|---|---|
| Slope is rise ÷ run between two points | In calculus, slope at a single point is called a derivative |
| y = mx + b describes a straight line | Systems of linear equations solve for where two lines cross |
| Similar triangles prove slope is constant | Trigonometry uses ratios in right triangles (sin, cos, tan) |
| Lines have equations with x to the first power | Quadratics have x² and create parabolas instead of lines |
You will also use slope-intercept form in science classes. In physics, a distance-time graph is a line when speed is constant, and the slope represents speed. In biology, you might graph population growth as a line and find its equation. The skills you build here will follow you everywhere.
Try these five problems to test your understanding. They go from easy to challenging. Give each one a try before reading the answer!
In this lesson you learned that the slope of a line is the ratio of rise to run. Using similar triangles, you proved that this ratio is the same between any two points on a non-vertical line. The Angle-Angle rule shows that slope triangles formed along a line are always similar, which is why the slope m never changes.
You then derived two key equations. For a line through the origin, the equation is y = mx. For a line with y-intercept b, the equation is y = mx + b. Remember: m tells you the steepness, b tells you where the line crosses the y-axis, and similar triangles are the reason the slope formula works everywhere on the line.