MathematicsChapter 8: Geometry of Straight Lines
Gradient As Tan
Gradient m = rise/run = tan theta.
Learning objectives
- Define and explain gradient as tan.
- Apply gradient as tan to a focused FBISE question.
- Check the answer for correct terms, evidence, working, and units.
Definition
Gradient m = rise/run = tan theta. Rise is the change in y; run is the change in x. From two points use m = y2-y1x2-x1. If the run is 0, you must not divide: that is the vertical case. If the rise is 0, m=0 and the line is horizontal. tan theta matches this because the slope triangle against the x-axis has opposite = rise and adjacent = run.
Mathematical form
m = rise/run
m = y2-y1x2-x1. If the run is 0
m=0 and the line is horizontal. tan theta
opposite = rise and adjacent
Where / symbols
- Gradient As Tan
- Gradient m = rise/run = tan theta. Rise is the change in y; run is the change in x. From two points use m = (y2-y1)/(x2-x1). If the run is 0, you must not divide: that is the vertical case. If the rise is 0, m=0 and the line is horizontal. tan theta matches this because the slope triangle against the x-axis has opposite = rise and adjacent = run.
Relationship
- m = rise/run
- m = y2-y1x2-x1
Overview
Learn the idea, then use it:
Core Understanding
Gradient m = rise/run = tan theta. Rise is the change in y; run is the change in x. From two points use m = y2-y1x2-x1. If the run is 0, you must not divide: that is the vertical case. If the rise is 0, m=0 and the line is horizontal. tan theta matches this because the slope triangle against the x-axis has opposite = rise and adjacent = run.
Examples m = rise/run · tan theta · m = (y2-y1)/(x2-x1) · GRADIENT AS TAN: Gradient m = rise/run = tan theta. Rise is the change in y; run is the change in x. From two points use m = (y2-y1)/(x2-
Exam Application
State the correct term, connect it to the chapter rule or evidence, and use a clear example or calculation.
Examples Define first · Show the rule or evidence · Check units, labels, and conclusion
WHAT — GRADIENT AS TAN
Gradient m = rise/run = tan theta. Rise is the change in y; run is the change in x. From two points use m = y2-y1x2-x1. If the run is 0, you must not divide: that is the vertical case. If the rise is 0, m=0 and the line is horizontal. tan theta matches this because the slope triangle against the x-axis has opposite = rise and adjacent = run.
HOW — Use the idea
Identify what is given, select this topic's rule or evidence, apply it one step at a time, and check the result.
In simple words
Do not merge gradient as tan with a neighbouring idea that uses a different definition, rule, or method.
Key points
- m = rise/run
- tan theta
- m = y2-y1x2-x1
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Gradient As Tan
Sticky Notes · Quick Revision
Core Idea
- Gradient m = rise/run = tan theta.
Key Points
- m = rise/run
- tan theta
- m = y2-y1x2-x1
Simple Check
- Explain gradient as tan 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
- Do not merge gradient as tan with a neighbouring idea that uses a different definition, rule, or method.
FBISE Exam Tip
- Use the exact term, show working where needed, and include units or labelled evidence.
60-second revision
- Gradient m = rise/run = tan theta.
- m = rise/run
- tan theta
- m = y2-y1x2-x1
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