📘 Remember
- Label the triangle before choosing a formula.
- Use degree mode unless the question says otherwise.
- Keep full calculator values until the final rounding step.
🎯 Learning Objectives
- derive and use Pythagoras’ theorem;
- distinguish side-only and angle-based problems;
- choose between Pythagoras and trigonometry;
- combine both methods in multi-step problems.
Introduction
This lesson develops Pythagoras or Trigonometry? from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Pythagoras’ Theorem
For a right triangle: a²+b²=c², where c is the hypotenuse.
Decision Flow
Is the triangle right-angled?
↓
Are you using only side lengths?
↓
Yes → Pythagoras
No, an acute angle and side are involved → Trigonometry
Worked Examples
Side-only problem
Legs 6 cm and 8 cm.
c²=6²+8²=100
c=10 cm.
Angle-and-side problem
Angle 40°, adjacent 8 cm, find opposite.
tan40°=x/8
x=8tan40°≈6.71 cm.
Practice Questions
- Find the hypotenuse when the legs are 5 cm and 12 cm.
- Find the missing leg when hypotenuse=13 cm and other leg=5 cm.
- Which method should be used when angle=37° and one side is known?
- A triangle has two known sides and no acute angle. Which method should be checked first?
✅ Check Your Work
Show answers and working
- c²=25+144=169, so c=13 cm.
- x²=13²−5²=144, so x=12 cm.
- Trigonometry.
- Pythagoras, provided it is a right triangle.
🧭 Interactive Lab: Choose the Method
Change the values or make a choice and watch the mathematics respond.
Select the best method for each situation.
Choose a situation, then choose the method.
Summary
⚠️ Common Mistakes
- Using the wrong reference angle.
- Rounding too early.
- Forgetting units or the degree symbol.
⭐ Exam Tips
- Write the formula before substituting.
- Sketch and label any diagram that is not already provided.
- Check whether the answer is sensible for the shape.
🌍 Did You Know?
Surveyors, engineers, pilots and architects use trigonometry to calculate distances that are difficult or unsafe to measure directly.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.