Lesson 1: Perimeter and Length – Pythagoras, Sine & Cosine

👨🏽‍🏫 Introduction

Have you ever walked diagonally across a football field to save time instead of walking along the sides? That’s Pythagoras in action! Or maybe you’ve seen surveyors measuring land with strange-looking tools — chances are, they’re using trigonometry to figure out distances without walking every step.

In this lesson, we’ll explore how to calculate the lengths and perimeters of 2D shapes using three powerful tools: the Pythagorean Theorem, the Sine Rule, and the Cosine Rule.

• 📏 Use the Pythagorean Theorem for right-angled triangles
• 📐 Use the Sine and Cosine Rules for general triangles (not just right-angled)
• 🌍 Apply these in real-life contexts like land measurement, architecture, and navigation

🧠 Key Concepts and Formulas

📌 1. Pythagorean Theorem (Right-Angled Triangles)

In any right-angled triangle:

, where c is the hypotenuse (longest side), and a and b are the shorter sides.

📌 2. Sine Rule (For Non-Right Triangles)

– use when you know:

• Two angles and one side (AAS or ASA)
• Two sides and an angle opposite one of them (SSA)

📌 3. Cosine Rule (For Non-Right Triangles)

– use when you know:

• Two sides and the included angle (SAS)
• All three sides (SSS) – to find any angle

🔍 Worked Examples

1. 1. Using Pythagoras: A ladder leans against a wall. The base is 6 m from the wall, and the ladder is 10 m long. How high does the ladder reach on the wall?

      Step 1: Use , with c = 10, a = 6.
      Step 2:
      Step 3:
      🟢 The ladder reaches 8 m high.

1. 1. Using the Sine Rule: In triangle ABC, angle A = 40°, angle B = 60°, and side a = 12 cm. Find side b.

      Step 1: First, find angle C = 180° – 40° – 60° = 80°.
      Step 2: Use the Sine Rule:
      →
      Step 3: Cross-multiply:
      🟢 Side b is approximately 16.17 cm.

1. Using the Cosine Rule: Triangle PQR has sides PQ = 7 cm, QR = 9 cm, and angle Q = 60°. Find side PR.

   Use the formula:
   
   
   
   🟢 Side PR is approximately 8.19 cm.

✍🏽 Practice Exercises

🅰️ Section A – Pythagorean Practice

1. A triangle has sides of 9 cm and 12 cm. Find the hypotenuse.
2. A flagpole casts a shadow 5 m long. The pole is 12 m tall. How far is the tip of the shadow from the top of the pole?

🅱️ Section B – Sine Rule

1. In triangle XYZ, angle X = 50°, angle Y = 70°, and side x = 14 cm. Find side y.

🅾️ Section C – Cosine Rule

1. In triangle LMN, sides LM = 8 cm, MN = 6 cm, and angle M = 90°. Find side LN.
2. In triangle ABC, AB = 10 cm, AC = 7 cm, and angle A = 45°. Find side BC.

✅ Answers and Explanations

• A1: cm
• A2: m
• B1: Angle Z = 60°
  → cm
• C1: cm
• C2: cm

📌 Recap

• ✅ Use Pythagoras for right-angled triangle sides
• ✅ Use Sine Rule when you have two angles and a side (or angle opposite a known side)
• ✅ Use Cosine Rule when you know two sides and the included angle or all three sides

💭 Reflection Prompt

Think about a place in real life where you’ve seen a triangle — maybe a ramp, a roof, or a road sign. How would you use one of today’s formulas to calculate an unknown length? Draw it out and try!