Slope in Geometry Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Slope in Geometry.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The steepness of a line expressed as rise over run, connecting the algebraic slope formula to the geometric angle of inclination.

A ramp's steepness—the ratio of how high it rises to how far it goes horizontally.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Slope = \frac{\text{rise}}{\text{run}} = \tan(\theta). Geometry and algebra connect here.

Common stuck point: Vertical lines have undefined slope (infinite steepness); horizontal lines have slope exactly zero.

Sense of Study hint: Pick two points on the line and compute rise over run. If the run is zero, the line is vertical with undefined slope.

Worked Examples

Example 1

easy
Find the slope of the line through A(-2, 5) and B(4, -1). Describe what the slope tells us about the line's direction.

Solution

  1. 1
    Step 1: Apply the slope formula: m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-1 - 5}{4 - (-2)} = \dfrac{-6}{6} = -1.
  2. 2
    Step 2: A slope of -1 means the line falls 1 unit for every 1 unit moved to the right.
  3. 3
    Step 3: The line makes a 45° angle below horizontal (since |m|=1 and the slope is negative, it descends left-to-right).

Answer

m = -1; the line descends at 45° from left to right.
Slope measures steepness and direction: positive slope rises left-to-right, negative slope falls, zero is horizontal, undefined is vertical. A slope of -1 means a 45° downward inclination, a special case where the line is perpendicular to slope +1 lines.

Example 2

hard
A road rises 40 metres over a horizontal distance of 500 metres. Express the slope as a percentage grade and as an angle (to the nearest tenth of a degree).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Which of these lines is steeper: y = 5x + 2 or y = -7x - 3? Which has a greater angle with the x-axis?

Example 2

medium
Points A(1, 2), B(4, k), C(7, 14) are collinear. Find k.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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