MathematicsChapter 8: Geometry of Straight Lines
Angle Between Lines and Families
If two lines have slopes m1 and m2, the tangent of the angle theta between them is |(m2-m1)/(1+m1 m2)| when 1+m1 m2 != 0.
Definition
If two lines have slopes m1 and m2, the tangent of the angle theta between them is |m2-m11+m1 m2| when 1+m1 m2 != 0. If 1+m1 m2 = 0 they are perpendicular (already a page). A family of lines shares a form with a parameter, such as y = mx + 2 (all slopes, same intercept 2).
Mathematical form
m2 = 0 they are perpendicular (already a page
y = mx + 2 (all slopes
Relationship
- tan theta = |m2-m11+m1 m2|
Overview
Angle, then the family:
Angle
Find m from each equation (make y = mx+c). Plug into the formula. If the lines are parallel, m1 = m2 and theta = 0. Acute or obtuse: the formula with absolute value gives the acute angle; the other is 180 deg minus that.
Examples m=1 and m=0 -> 45 deg · Vertical line: slope undefined, handle separately · Do not use m as an angle
Families
y = 2x + k is all parallels of slope 2. y - 3 = m(x-1) is all lines through (1,3). The pair of axes is a special family. A question 'member of the family that...' means substitute the condition to find the parameter.
Examples Through (1,3) and slope 4 · Parallel to y=3x-1 through origin · k from a second given point
Explanation
Point-slope and two-point forms were earlier. The family is those forms with one letter left free.
In simple words
If 1+m1 m2 is zero, stop and write 'perpendicular'; do not divide by zero in the tan formula.
Key points
- tan theta = |m2-m11+m1 m2|
- parameter in a family
- make y = mx+c first
- k slides a parallel
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Angle Between Lines and Families
Sticky Notes · Quick Revision
Core Idea
- If two lines have slopes m1 and m2, the tangent of the angle theta between them is |m2-m11+m1 m2| when 1+m1 m2 != 0.
Key Points
- tan theta = |m2-m11+m1 m2|
- parameter in a family
Simple Check
- Explain angle between lines and families in your own words and connect it to one chapter example.
How to Answer
- Define the idea, show the main rule or evidence, apply it, and check the final statement.
Common Mistake
- If 1+m1 m2 is zero, stop and write 'perpendicular'; do not divide by zero in the tan formula.
FBISE Exam Tip
- Use the exact term, show working where needed, and include units or labelled evidence.
60-second revision
- tan theta = |m2-m11+m1 m2|
- parameter in a family
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