Mountain trails, wheelchair ramps, ski slopes, horse trailer loading ramps — all of these have a steepness. That steepness is measured with a single number: slope. And thanks to what you now know about similar triangles, you can understand why that number never changes along a straight path.
Why slope is constant — the similar triangles argument:
Pick any two rise/run right triangles under the same straight line. Both triangles share the same angle where the line meets the horizontal. Two equal angles means AA similarity — so the triangles are similar. Similar triangles have proportional sides. Therefore:
$$\frac{\text{rise}_1}{\text{run}_1} = \frac{\text{rise}_2}{\text{run}_2} = \text{constant}$$
That constant ratio is the slope $m$:
$$m = \frac{\text{rise}}{\text{run}} = \frac{\text{vertical change}}{\text{horizontal change}}$$
Positive slope → line goes up left-to-right. Negative → down. Zero → flat. Undefined → vertical.
Similar Triangles Argument — Visual
Worked Example 1 — Two triangles, same slope
A straight path climbs $3$ m up for every $4$ m across (small triangle). A different section of the same path climbs $6$ m up for every $8$ m across (big triangle). Are the triangles similar? What is the slope?
Small triangle: rise $= 3$, run $= 4$. Big triangle: rise $= 6$, run $= 8$.
Ratio small: $\frac{3}{4} = 0.75$. Ratio big: $\frac{6}{8} = 0.75$ ✓ — equal ratios confirm similarity.
$$m = \frac{3}{4} = 0.75$$
The slope is the same everywhere on the path — because the triangles are always similar.
Worked Example 2 — Read slope from a described grid
A line passes through the origin and through the point $(5, 2)$. Count the rise and run.
From $(0,0)$ to $(5,2)$: run $= 5$ (across), rise $= 2$ (up).
$$m = \frac{2}{5} = 0.4$$
The line rises gently — a $40\%$ grade, which is too steep for a horse trail (safe max ≈ $15\%$).
Warm-Up — Read slope from rise and run (values given)
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Problem 1
A stable ramp rises $1.2$ m over a horizontal distance of $6$ m. Find the slope. Is this safe for horses? (Max safe slope for horses: about $0.15$ or $15\%$.)
Equation: $m = \dfrac{\text{rise}}{\text{run}} = \dfrac{1.2}{6}$
calculate → compare to 0.15 → safe / not safe →
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Problem 2
A hiking trail in Bavaria drops $80$ m in altitude over a horizontal distance of $600$ m. Find the slope. Is it positive or negative? Why?
Equation: $m = \dfrac{-80}{600}$
calculate → state sign → explain →
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Problem 3
A perfectly flat paddock has zero elevation change. What is the slope of the paddock surface? What does that look like on a graph?
calculate → describe graph →
Core Problems — interpret slope in real situations
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Problem 4
A horse trailer loading ramp must not exceed a $20\%$ grade ($m = 0.20$). The ramp needs to reach a trailer floor that is $0.9$ m off the ground. What is the minimum horizontal length the ramp must be? (Hint: rearrange $m = \text{rise}/\text{run}$.)
rearrange slope formula → solve for run →
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Problem 5
Mia plots two points from a walk: she starts at a height of $240$ m and walks forward $500$ m horizontally to reach $270$ m. Find the slope. Then: a second trail rises $270$ m to $330$ m over only $300$ m horizontal. Which trail is steeper?
find slope of each trail → compare →
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Problem 6
In the arithmetic sequence $3, 7, 11, 15, \ldots$ the term number is $x$ and the value is $y$. The table below shows some values. If you plot term number vs. value, what is the slope of the line? How does this connect to the common difference?
| $x$ (term number) | $y$ (value) |
|---|
| 1 | 3 |
| 2 | 7 |
| 3 | 11 |
| 4 | 15 |
calculate rise/run between consecutive points → state slope → connect to common difference →
From here — write the equation yourself, then solve
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Problem 7
A ramp is to be built from the street to a door $0.45$ m above street level. Building code requires slope $\leq 0.125$ (i.e., $12.5\%$). What is the minimum ramp length you need (horizontal distance)?
write slope formula → rearrange → solve →
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Problem 8 Challenge
Two lines pass through the same point $(0, 4)$. Line A also passes through $(3, 10)$. Line B also passes through $(6, 4)$.
(a) Find the slope of each line.
(b) Describe in words what each line looks like.
(c) Which slope has a larger absolute value? What does that mean about steepness?
calculate slopes → describe → compare absolute values →
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Problem 9 Open
Design a horse trail on a hillside that is challenging but safe. Your trail must: (1) have a slope between $0.08$ and $0.15$; (2) gain at least $200$ m in altitude; (3) be as short as possible horizontally. Describe your trail with specific numbers and explain why this slope is appropriate for horses.
choose slope → rearrange for run → compute → justify →
Show answers
Problem 1
$m = 1.2 / 6 = 0.2$ · This is $20\%$, which exceeds the safe maximum of $15\%$ — not safe for horses.
Problem 2
$m = -80/600 \approx -0.133$ · Negative because the trail goes down (altitude decreases as horizontal distance increases).
Problem 3
$m = 0$ · On a graph it appears as a horizontal (flat) line.
Problem 4
$m = \text{rise}/\text{run}$ → $\text{run} = \text{rise}/m = 0.9/0.20 = 4.5$ m · The ramp must be at least $4.5$ m long horizontally.
Problem 5
Trail 1: $m = (270-240)/500 = 30/500 = 0.06$ · Trail 2: $m = (330-270)/300 = 60/300 = 0.20$ · Trail 2 is steeper ($0.20 > 0.06$).
Problem 6
Rise between consecutive terms $= 4$, run $= 1$, so $m = 4/1 = 4$ · This equals the common difference! The slope of a plotted arithmetic sequence always equals the common difference.
Problem 7
$\text{run} = 0.45/0.125 = 3.6$ m minimum horizontal length.
Problem 8
(a) Line A: $m = (10-4)/(3-0) = 6/3 = 2$ · Line B: $m = (4-4)/(6-0) = 0/6 = 0$ · (b) Line A is steep and rising; Line B is perfectly flat. (c) $|2| > |0|$ — Line A is steeper.
Problem 9
Open — any trail with slope in $[0.08, 0.15]$ and altitude gain $\geq 200$ m is valid. Example: slope $= 0.10$, run $= 200/0.10 = 2\,000$ m horizontal. The slope is gentle enough for horses to walk comfortably without straining their hooves or joints.
Coming up next → Lesson 17: Slope from Two Points
You can now read slope from a grid or a described situation. But what if you only know two coordinates — no grid? Lesson 17 introduces the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ and shows you how to use it for any two points.