Mathematics
Grade 10
15 min
Slopes of lines
Slopes of lines
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1
Introduction & Learning Objectives
Learning Objectives
Calculate the slope of a line given two points on the line.
Interpret the meaning of positive, negative, zero, and undefined slopes in the coordinate plane.
Determine if two lines are parallel, perpendicular, or neither by comparing their slopes.
Use the slope formula to find a missing coordinate of a point on a line.
Write a proof to show that three or more points are collinear using the concept of slope.
Apply the concept of slope to solve problems involving geometric figures like triangles and quadrilaterals in the coordinate plane.
Have you ever wondered why some ski slopes are labeled 'Beginner' while others are 'Expert'? ⛷️ It all comes down to a mathematical concept called slope!
This tutorial will explore the fundamental conce...
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Key Concepts & Vocabulary
TermDefinitionExample
SlopeA number that measures the 'steepness' and direction of a non-vertical line. It is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.If a line passes through (1, 2) and (3, 6), the rise is 4 and the run is 2, so the slope is 4/2 = 2.
RiseThe vertical change between two points on a line. It is calculated as the difference in their y-coordinates (y₂ - y₁).For the points (2, 5) and (4, 12), the rise is 12 - 5 = 7.
RunThe horizontal change between two points on a line. It is calculated as the difference in their x-coordinates (x₂ - x₁).For the points (2, 5) and (4, 12), the run is 4 - 2 = 2.
Positive SlopeA line that goes upward from left to right. This occurs when both the rise and run are positive o...
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Core Formulas
The Slope Formula
Given two points (x₁, y₁) and (x₂, y₂), the slope 'm' is: m = \frac{y_2 - y_1}{x_2 - x_1}
Use this formula to calculate the slope of a line when you know the coordinates of any two points on that line. It represents the 'rise over run'.
Slopes of Parallel Lines
Two distinct non-vertical lines are parallel if and only if their slopes are equal. m₁ = m₂
If you calculate the slopes of two different lines and find that the slopes are identical, the lines are parallel. They will never intersect.
Slopes of Perpendicular Lines
Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. m₁ \cdot m₂ = -1
This means their slopes are negative reciprocals of each other (e.g., 2/3 and -3/2). Use this rule to...
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Challenging
The vertices of a quadrilateral are A(0,0), B(2,4), C(8,1), and D(6,-3). By analyzing the slopes of its sides and diagonals, what is the most specific classification for this figure?
A.Parallelogram
B.Rhombus
C.Rectangle
D.Square
Challenging
Find the value of 'k' such that the line through (k, 4) and (-2, 1) is perpendicular to the line through (-5, 3) and (-8, k).
A.3
B.0
C.2
D.4
Challenging
Three points A(1, 5), B(4, y), and C(x, -1) are collinear. If the slope of the line segment AB is -2, what are the values of x and y?
A.x = 3, y = -1
B.x = 4, y = -1
C.x = 4, y = 1
D.x = -1, y = -1
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