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Lesson 1 of 5

Straight Lines · Lesson 1 of 5

Slope and Relationships Between Lines

“Connect coordinate geometry to the direction of a line, then use slope to explain parallelism, perpendicularity and angles.”

Learning Objectives

• Use distance, section, midpoint and triangle-area formulas in coordinate geometry. • Interpret inclination and signed slope, including horizontal and vertical lines. • Derive slope from two points and explain why it stays constant. • Use slope to establish parallelism, perpendicularity and collinearity. • Find angles between lines and recognise when two directions are possible.

Coordinates Turn Geometry into Algebra

A point is a location; its coordinates tell us how to reach that location from the origin. A line is a whole collection of points with a shared geometric relationship. Our aim is to describe that relationship using equations, beginning with the direction in which the line runs.

In a Cartesian plane, the x-axis is horizontal and the y-axis is vertical. The ordered pair (x, y) gives a signed horizontal displacement followed by a signed vertical displacement. The point (6, −4) is six units to the right and four units below the origin, while (3, 0) lies on the x-axis. Coordinates can be negative; a distance or area cannot be negative.

Coordinate geometry combines algebra with geometry. Descartes helped establish this approach in the seventeenth century: a geometric condition can become an equation, and solving an equation can locate a point. Before studying lines, recall four tools that let us measure and divide figures.

Distance between two pointsLaTeX
P=(x₁,y₁) and Q=(x₂,y₂). The horizontal and vertical coordinate differences are the perpendicular sides of a right triangle; PQ is its hypotenuse.

The formula follows from the Pythagorean relationship, so the order of subtraction does not change the result. If both x-coordinates agree, the horizontal change is zero and the distance is |y₂−y₁|. If both y-coordinates agree, the distance is |x₂−x₁|. This also explains why distance is different from a signed coordinate change.

Internal division of a segmentLaTeX
R divides PQ internally in the ratio PR:RQ=r:s, with r,s>0. The weight r multiplies Q’s coordinates, not P’s.

Internal division means R lies between P and Q. If PR is a small fraction of the whole segment, R must be close to P. This is a useful way to check the placement of the weights. For A=(1,−3), B=(−3,9), and AR:RB=1:3, the formula gives R=((−3+3)/4,(9−9)/4)=(0,0).

MidpointLaTeX
Set the division ratio to 1:1. Each midpoint coordinate is the average of the corresponding endpoint coordinates.
Triangle areaLaTeX
The three vertices are (x₁,y₁), (x₂,y₂), (x₃,y₃). The absolute value makes the area nonnegative. Zero area means the three points are collinear.
Example — Coordinate measurements

Problem
Find the distance between (6,−4) and (3,0), then the area of the triangle with vertices (4,4), (3,−2), (−3,16).

  1. 1.The coordinate changes for the segment are 3−6=−3 and 0−(−4)=4.
  2. 2.Its length is √(9+16)=5 units.
  3. 3.For the area, substitute in cyclic order: ½|4(−2−16)+3(16−4)+(−3)(4−(−2))|.
  4. 4.The expression inside the absolute value is −72+36−18=−54, so the area is 27 square units.

Inclination and Slope

A line’s direction can be described by an angle or by a ratio of coordinate changes. The angle gives a visual direction; the ratio is usually easier to calculate. Both descriptions tell us how the line rises or falls as we move across the plane.

Definition
Inclination

The angle θ measured anticlockwise from the positive x-direction to a line. Choose 0°≤θ<180°, with a horizontal line assigned 0° and a vertical line assigned 90°. The direction is for the entire line, not just a ray pointing towards one endpoint.

Slope from inclinationLaTeX
m is the slope, also called the gradient. A horizontal line has m=0; a vertical line has no defined finite slope.

A positive slope means the line rises as x increases; its inclination is acute. A negative slope means the line falls as x increases; its inclination is obtuse. The magnitude describes steepness relative to the chosen coordinate scales: a slope of 2 means a vertical increase of two coordinate units for a horizontal increase of one. Slope is a ratio, not a distance.

Take two distinct points P=(x₁,y₁) and Q=(x₂,y₂) on a nonvertical line. Choose their order so x₂>x₁ and draw a horizontal segment from P to the vertical through Q. For a rising line, the right triangle has horizontal side x₂−x₁ and vertical side y₂−y₁. Thus tan θ is vertical change divided by horizontal change.

For a falling line, let φ=180°−θ be the acute angle in the right triangle. Its positive vertical side length is y₁−y₂, so tan φ=(y₁−y₂)/(x₂−x₁). Since tan θ=−tan φ, the signed numerator becomes y₂−y₁. Therefore the same signed formula works for both acute and obtuse inclinations.

Slope through two pointsLaTeX
Keep the subtraction order consistent. Reversing both orders multiplies numerator and denominator by −1 and leaves their ratio unchanged.
Signed rise and runxyOP (−2,1)Q (4,4)R (4,1)
Signed rise and run— P=(−2,1), Q=(4,4). The horizontal run is 6 and the rise is 3, so the slope is 1/2.

The ratio is constant along a straight line because any two of these right triangles have the same inclination and are similar. This is what makes a slope a property of the line rather than a property of one chosen segment. If the two chosen points have the same x-coordinate, the denominator is zero: the line is vertical and must be handled separately.

Example — Four kinds of direction

Problem
Find slopes through (3,−2) and (−1,4), through (3,−2) and (7,−2), through (3,−2) and (3,4), and for inclination 60°.

  1. 1.For the first pair, m=(4+2)/(−1−3)=6/(−4)=−3/2. The line falls to the right.
  2. 2.For the second pair, m=(−2+2)/(7−3)=0/4=0: a horizontal line.
  3. 3.For the third pair, the denominator is 3−3=0. Its slope is undefined, and its equation will be x=3.
  4. 4.For the angle, m=tan 60°=√3. Here the angle gives direction directly.
Common mistake

Undefined slope does not mean zero slope. Zero slope describes a horizontal line; undefined slope describes a vertical line. A line’s inclination is measured from the positive x-axis. If an angle is given from the positive y-axis, convert the reference direction first.

Example — An angle measured from the y-axis

Problem
A line makes 30° anticlockwise with the positive y-direction. Find its slope.

  1. 1.The positive y-direction is already 90° anticlockwise from the positive x-direction.
  2. 2.Therefore θ=90°+30°=120°.
  3. 3.m=tan 120°=−√3. The negative sign agrees with an obtuse inclination.

Parallel, Perpendicular and Collinear

We can compare the directions of lines without measuring every angle or every side. Parallel lines share a direction. Perpendicular lines differ in direction by a right angle. Collinear points all lie along one direction.

For two nonvertical lines, equal inclinations give equal tangents and therefore equal slopes. Conversely, tangent determines a unique inclination in our chosen interval apart from the excluded 90°, so equal slopes give the same direction. Two distinct lines of equal slope are parallel; if they also share a point they are the same line. Two vertical lines likewise have the same direction.

Parallelism for nonvertical linesLaTeX
Use this for direction. To distinguish coincident from distinct parallel lines, also compare a point or the intercepts.

If the inclinations differ by 90°, the tangent identity tan(α+90°)=−1/tan α gives the slope of the perpendicular direction. This explains the negative reciprocal: a line rising steeply is perpendicular to a line falling gently. This calculation assumes both slopes are finite; horizontal and vertical lines form the remaining perpendicular pair.

Perpendicularity for finite slopesLaTeX
For nonzero finite m₁, the perpendicular slope is m₂=−1/m₁. A horizontal line is perpendicular to a vertical line.
Example — Find an unknown coordinate

Problem
The line through (−2,6) and (4,8) is perpendicular to the line through (8,12) and (x,24). Find x.

  1. 1.The first slope is (8−6)/(4+2)=1/3.
  2. 2.The required second slope is −3. The second pair has slope 12/(x−8), provided x≠8.
  3. 3.Set 12/(x−8)=−3. Multiply by x−8 to obtain 12=−3x+24.
  4. 4.Thus x=4. Check: 12/(4−8)=−3, and (1/3)(−3)=−1.
Example — A right angle without side lengths

Problem
Show that A=(4,4), B=(3,5) and C=(−1,−1) form a right-angled triangle.

  1. 1.The slope of AB is (5−4)/(3−4)=−1.
  2. 2.The slope of AC is (−1−4)/(−1−4)=1.
  3. 3.Their product is −1, so AB is perpendicular to AC. The right angle is at A.
  4. 4.The slopes are different, so the points are not collinear; they form a genuine triangle.

For collinearity, compare slopes of segments joining the points when those slopes exist. If all x-coordinates are equal, the points lie on a vertical line, and no slope calculation is needed. For a parallelogram, show that each pair of opposite sides has equal direction; also check that adjacent sides are not collinear. For polygons, split the region into triangles and add their areas rather than treating a four-vertex figure as one triangle.

Angles Between Lines

Two intersecting lines create two equal acute angles and two equal obtuse angles, unless the lines are perpendicular. A slope calculation must distinguish the smaller angle from an oriented difference of inclinations. An absolute value is what turns that signed difference into the smaller nonnegative angle.

Let the inclinations be α₁ and α₂. Applying tan(α₂−α₁)=(tan α₂−tan α₁)/(1+tan α₁ tan α₂) gives (m₂−m₁)/(1+m₁m₂). The signed result may describe an obtuse angle. For the smaller angle θ, take its absolute value. If the denominator is zero, the lines are perpendicular and θ=90° instead of being found by division.

Smaller angle between finite-slope linesLaTeX
For nonparallel, nonperpendicular lines, 0°<θ<90°. The other angle is 180°−θ. Equal slopes give θ=0°.

If one line is vertical, use its 90° inclination directly and compare it with the other line’s inclination. For a vertical line and a line of nonzero finite slope m, tan θ=1/|m|. A vertical and a horizontal line meet at 90°. These cases are part of the geometry, even though the two-finite-slope formula cannot accept an undefined slope.

Example — Why an angle can give two slopes

Problem
A line of slope 1/2 makes a smaller angle of 45° with another line. Find the other slope.

  1. 1.Let the other slope be m. Then |(m−1/2)/(1+m/2)|=tan 45°=1.
  2. 2.The positive branch gives m−1/2=1+m/2, hence m/2=3/2 and m=3.
  3. 3.The negative branch gives m−1/2=−1−m/2, hence 3m/2=−1/2 and m=−1/3.
  4. 4.Both denominators are nonzero. Substitution gives tangent magnitude 1 in each case.
  5. 5.Geometrically, the unknown direction can lie on either side of the known direction, which explains two answers.
Example — Coordinate construction of an equilateral triangle

Problem
An equilateral triangle of side 2a, a>0, has its base on the y-axis with midpoint at the origin. Find all possible vertices.

  1. 1.The base endpoints are A=(0,a), B=(0,−a), because the origin bisects a vertical segment of length 2a.
  2. 2.The third vertex C lies on the perpendicular bisector of AB, which is the x-axis. Write C=(t,0).
  3. 3.The condition AC=2a gives t²+a²=4a², so t²=3a².
  4. 4.Thus C=(√3a,0) or C=(−√3a,0). Both triangles satisfy the construction and lie on opposite sides of the base.
Example — Area of a coordinate quadrilateral

Problem
Find the area of the quadrilateral with successive vertices A=(−4,5), B=(0,7), C=(5,−5), D=(−4,−2).

  1. 1.Sketch the points in the stated order and split the convex quadrilateral along AC into triangles ABC and ACD.
  2. 2.Twice the area of ABC is |(0−(−4))(−5−5)−(7−5)(5−(−4))|=|4(−10)−2(9)|=58. Thus its area is 29.
  3. 3.Twice the area of ACD is |(5−(−4))(−2−5)−(−5−5)(−4−(−4))|=|9(−7)−(−10)(0)|=63. Thus its area is 63/2.
  4. 4.The total is 29+63/2=121/2 square units. The diagonal subdivision ensures each part is counted once.

Quiz

Quick check

Which pair describes a vertical line?

Quick check

The slope through (−1,2) and (3,−6) is:

Quick check

A nonvertical line perpendicular to slope −3 has slope:

Quick check

The smaller angle between slopes 1 and 0 is:

Quick check

Equal finite slopes prove that two lines:

Quick check

A triangle has zero coordinate area. What follows?

Key Takeaways

Key Takeaways

• Slope is signed vertical change divided by signed horizontal change; it equals the tangent of inclination. • Horizontal slope is zero; vertical slope is undefined. • Equal finite slopes give equal directions; negative reciprocal finite slopes give perpendicular directions. • Use absolute values for distance, area and the smaller-angle tangent. • Two possible slopes can arise from the two angle directions; check both. • Coordinate distance, internal division, midpoint and area support line geometry.

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Equations of Straight Lines