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: A line's slope equals both $\frac{\text{rise}}{\text{run}}$ and $\tan\theta$ , where $\theta$ is the angle it makes with the $x$ -axis.

Common stuck point: The procedure for slope in geometry is the easy part; the trap is reading the slope value as the angle in degrees. Asking "Am I connecting a line's rise-over-run to the angle it makes with the horizontal?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I connecting a line's rise-over-run to the angle it makes with the horizontal?

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.

Line through A(−2, 5) and B(4, −1)

Answer

m = -1

; the line descends at

45°

from left to right.

First step

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

Full solution

2
Step 2: A slope of $-1$ means the line falls $1$ unit for every $1$ unit moved to the right.
3
Step 3: The line makes a $45°$ angle below horizontal (since $|m|=1$ and the slope is negative, it descends 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).

Example 3

easy

Find the slope of the line through

(-5, 2)

and

(3, -2)

Find the slope of the line through (−5, 2) and (3, −2)

Example 4

medium

A line has slope

\tfrac{3}{5}

and passes through

(2, -1)

. Find its

y

-intercept.

Example 5

medium

A line through the origin makes an angle of

60°

with the positive

x

-axis. Find its slope (exact value).

Line through origin at 60° inclination — find its slope

Example 6

medium

A roof rises

5

ft over a horizontal run of

12

ft. What is the angle of inclination to the nearest degree?

Example 7

medium

Find the slope of the line that is perpendicular to

2x - 5y = 10

Example 8

hard

Find the acute angle between the lines

y = 2x + 1

and

y = -3x + 4

to the nearest degree.

Find the acute angle between y = 2x + 1 and y = −3x + 4

Example 9

hard

Points

A(1, 2)

B(5, 6)

C(3, 8)

form a triangle. Find the slope of the altitude from

C

to side

AB

Example 10

hard

Find the slope of the line tangent to the circle

x^2 + y^2 = 25

at the point

(3, 4)

Example 11

challenge

A line through

(2, 3)

with negative slope

m

cuts off a triangle from the first quadrant with the coordinate axes. Find the slope

m

that minimizes the length of the segment between the two axis intercepts.

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

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

Three collinear points; find the missing y-coordinate k at x = 4

Example 3

easy

A line rises 6 for every run of 3. What is its slope?

Example 4

easy

What is the slope of a horizontal line?

Example 5

easy

A slope of 1 corresponds to what angle with the horizontal?

Example 6

easy

Find the slope of the line through

(1, 2)

and

(4, 8)

Example 7

easy

Does a steeper line have a larger or smaller absolute slope?

Example 8

easy

A line goes down from left to right. Is its slope positive or negative?

Example 9

easy

Why is the slope of a vertical line undefined?

Example 10

easy

A ramp rises 1 m over a horizontal run of 12 m. What is its slope?

Example 11

medium

A line makes a

60^\circ

angle with the horizontal. Find its slope (leave in exact form).

A line making a 60° angle with the positive x-axis

Example 12

medium

Two points on a line give slope 0. What does this tell you about the line?

Example 13

medium

A line has slope

-\tfrac{3}{4}

. If it rises (changes

y

) by

-6

, what is the change in

x

Example 14

medium

Why does a constant slope mean the line is straight?

Example 15

medium

A line passes through

(2, 5)

with slope 3. Find its

y

-intercept.

Example 16

medium

Points

A(1,2)

B(3,6)

C(5,10)

are given. Use slopes to check if they are collinear.

Points A(1,2), B(3,6), C(5,10) — are they collinear?

Example 17

medium

A road has a grade of 5% (rises 5 units per 100 horizontal). Express its slope as a decimal.

Example 18

medium

A line of slope 0.5 and a line of slope 5 are drawn. Which is steeper, and how does the angle compare?

Example 19

challenge

A line has slope 1. Another line is perpendicular to it. Find the angle each makes with the horizontal, and confirm they differ by

90^\circ

Lines with slopes 1 and −1 are perpendicular; verify the angles differ by 90°

Example 20

challenge

The points

(0, b)

(2, 5)

(4, 11)

are collinear. Find

b

Collinear points force the line through (0, b); find b

Example 21

challenge

A line through the origin has inclination angle

\theta

such that the line passes through

(4, 3)

. Find

\tan\theta

and

\sin\theta

Line through the origin and (4, 3) with its 3-4-5 right triangle

Example 22

challenge

Explain why slope is the same between ANY two points on a given line, and why this fails for a curve.

Example 23

easy

Find the slope of the line through

(2, 3)

and

(6, 11)

Example 24

easy

Find the slope of the line

3x + 4y = 12

Example 25

easy

What is the slope of a vertical line?

Example 26

easy

A wheelchair ramp rises

0.5

m over a run of

6

m. What is its slope?

Example 27

medium

The points

(2, k)

(4, 5)

, and

(8, 13)

are collinear. Find

k

Collinear points — find the missing coordinate k

Example 28

medium

A line has slope

-\tfrac{2}{3}

. A perpendicular line through

(1, 5)

has what equation?

Example 29

medium

The line

y = mx + 2

passes through

(-3, -7)

. Find

m

Example 30

medium

The median rent in a city rose from $1,200 in 2018 to $1,560 in 2022. Treating year as

x

and rent as

y

, find the slope (units $/year).

Example 31

medium

Given two lines

y = (k - 2)x + 3

and

y = (2k + 1)x - 1

, find

k

so they have the same slope.

Example 32

hard

A treadmill is inclined at

7.5\%

grade. What is the angle of incline to the nearest tenth of a degree?

Example 33

hard

Find the slope of the line through

(2, 3)

and the point on the positive

x

-axis at distance

5

from the origin.

Example 34

hard

Find the acute angle between

y = x

and

y = (2 + \sqrt{3}) x

in degrees.

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Related Concepts

Slope Angles Trigonometric Functions Tangent Intuition

Background Knowledge

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

slopeangles