Triangles • Topic 3 of 3

Pythagoras Theorem and its converse

What is the Pythagoras Theorem? The Pythagoras Theorem is one of the most famous and useful rules in geometry. It applies exclusively to right-angled triangles (triangles where one angle is exactly 90 degrees).

The theorem states: In a right-angled triangle, the square of the length of the hypotenuse (the longest side, directly opposite the 90-degree angle) is equal to the sum of the squares of the lengths of the other two sides (often called the base and the height).

Mathematical Equation:

$$\text{Base}^2 + \text{Height}^2 = \text{Hypotenuse}^2$$

Or simply:

$$a^2 + b^2 = c^2$$

(where $c$ is the length of the hypotenuse, while $a$ and $b$ are the other two sides).

What is the Converse of the Pythagoras Theorem? The word "converse" means looking at a rule backward. The Converse of the Pythagoras Theorem states: If a triangle has three sides such that the square of the longest side equals the sum of the squares of the other two sides, then the triangle must be a right-angled triangle. The 90-degree angle will always be located directly opposite that longest side.

---

DIAGRAM 1: THE RIGHT-ANGLED TRIANGLE STRUCTURE

         |\
         | \
         |  \   Hypotenuse (c) -> Longest side, opposite to 90°
  Side 1 |   \
     (a) |    \
         |_____\
          Side 2 (b)
          [Note the square box corner indicating the 90° angle]

DIAGRAM 2: VISUAL PROOF OF SQUARES (3-4-5 Triangle)

             /\
            /  \ 5
         4 /____\
            3
            
     Side 1 squared: 4 * 4 = 16 blocks
     Side 2 squared: 3 * 3 = 9 blocks
     Hypotenuse squared: 5 * 5 = 25 blocks
     Notice that: 16 + 9 = 25!

DIAGRAM 3: REAL-WORLD USE (LADDER ON A WALL)
         | \
    Wall |  \ Ladder (c)
     (a) |   \
         |____\
          Ground (b)
Solved Examples
7
Worked Example
A right-angled triangle has a base of 6 cm and a height of 8 cm. Find the length of its hypotenuse.
Solution
  1. Step 1: Identify the given values and what to find.*
  2. Base (a) = 6 cm, Height (b) = 8 cm. We need to find the Hypotenuse (c).*
  3. Step 2: Apply the Pythagoras Theorem formula.*
  4. * $$a^2 + b^2 = c^2$$
  5. * $$6^2 + 8^2 = c^2$$
  6. Step 3: Calculate the squares.*
  7. * $$36 + 64 = c^2$$
  8. * $$100 = c^2$$
  9. Step 4: Find the square root.*
  10. c = Square root of 100 = 10 cm.*
  11. Answer: The length of the hypotenuse is 10 cm.
8
Worked Example
A ladder is placed against a high vertical wall. The foot of the ladder is 5 meters away from the base of the wall, and the top of the ladder reaches a window that is 12 meters high up the wall. What is the length of the ladder?
Solution
  1. Step 1: Visualize the setup as a right triangle.*
  2. The wall and the ground form a 90-degree angle. The wall is the height (12 m), the ground distance is the base (5 m), and the ladder itself forms the hypotenuse.*
  3. Step 2: Apply the Pythagoras Theorem.*
  4. * $$\text{Ladder}^2 = \text{Wall}^2 + \text{Ground}^2$$
  5. * $$\text{Ladder}^2 = 12^2 + 5^2$$
  6. Step 3: Calculate the sum.*
  7. * $$\text{Ladder}^2 = 144 + 25$$
  8. * $$\text{Ladder}^2 = 169$$
  9. Step 4: Take the square root.*
  10. Ladder length = Square root of 169 = 13 meters.*
  11. Answer: The length of the ladder is 13 meters.
3
Worked Example
A triangle has side lengths of 7 cm, 24 cm, and 25 cm. Determine whether this triangle contains a right angle.
Solution
  1. Step 1: Identify the longest side.*
  2. The longest side is 25 cm. The other two sides are 7 cm and 24 cm.*
  3. Step 2: Calculate the square of the longest side.*
  4. * $$25^2 = 625$$
  5. Step 3: Calculate the sum of the squares of the remaining two sides.*
  6. * $$7^2 + 24^2 = 49 + 576$$
  7. * $$49 + 576 = 625$$
  8. Step 4: Check if they match using the Converse theorem.*
  9. Since the square of the longest side (625) equals the sum of the other two squares (625), the triangle satisfies the Converse of the Pythagoras Theorem.*
  10. Answer: Yes, it is a right-angled triangle.
  11. --

Key Points

  • The Pythagoras Theorem is exclusively true for right-angled triangles.
  • The fundamental formula is
  • The hypotenuse always sits directly opposite the 90-degree angle.
  • Common sets of whole numbers that fit this theorem perfectly are called Pythagorean triplets (e.g., 3-4-5, 5-12-13, 7-24-25).
  • The Converse rule allows you to prove an angle is exactly 90 degrees simply by measuring its three side lengths.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.In a right triangle, (hypotenuse)$^2$ equals:
Explanation: Pythagoras' theorem.
Q2.$3,4,5$ is a:
Explanation: $3^2+4^2=5^2$.
Q3.If $a^2+b^2=c^2$, the triangle is:
Explanation: Converse of Pythagoras.
Q4.The hypotenuse for legs $6$ and $8$ is:
Explanation: $\sqrt{36+64}=10$.
Practice more MCQs → 📝 Assignment · 35 marks