Grade 9 Pythagoras Relationship: Short Notes

9 ශ්‍රේණිය · 19 පරිච්ඡේදය · පයිතගරස් සම්බන්ධය

70%

of Grade 9 term tests include it (28 of 40)

195

marks in past papers (35 questions)

10 min

full lesson with quick checks

🎯 What you'll learn

Identify the hypotenuse of a right-angled triangle
Write Pythagoras' relationship for a labelled right triangle
Find the hypotenuse or a missing side using Pythagoras
Test whether three lengths form a right-angled triangle
Solve everyday and two-step problems using Pythagoras

The shortcut across the ground

Your school ground is a rectangle 40 m long and 30 m wide. To reach the canteen in the far corner you can walk along two sides, or cut straight across. How many metres does the shortcut save? A rule found over 2500 years ago tells you, without measuring.

Squares on the sides

In a right triangle with sides 3, 4 and 5, the squares on the two shorter sides have areas 9 and 16. Together they make 25, the area of the square on the longest side: 9 + 16 = 25.

🧮 Pythagoras' relationship

c is the hypotenuse: the side opposite the right angle, always the longest. a and b are the two sides that form the right angle. It works only in a right-angled triangle.

c² = a² + b²

✏️ Finding the hypotenuse

In triangle PQR, the angle at Q is 90°, PQ = 6 cm and QR = 8 cm. Find PR.

1PR is opposite the right angle, so it is the hypotenuse.
2PR² = PQ² + QR² = 6² + 8² = 36 + 64 = 100.
3PR = √100 = 10 cm.

✅ 10 cm

✏️ Finding a shorter side

A right triangle has hypotenuse 25 cm and one other side 7 cm. Find the third side.

125² = 7² + b², so b² = 25² − 7².
2b² = 625 − 49 = 576.
3b = √576 = 24 cm.

✅ 24 cm

⚠️ Add for the hypotenuse, subtract for the others

Two common slips:1. Adding squares when the hypotenuse is given. For 25 and 7 you get √674 ≈ 26, longer than the hypotenuse, which is impossible. Subtract instead.2. Stopping at b² = 576 and forgetting the square root.Check: the hypotenuse must be the longest side.

Is it a right angle? Test it

It also works backwards: if (longest side)² = sum of the squares of the other two, the triangle is right-angled.Whole-number sets that work are Pythagorean triples: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25. Multiples work too: 6, 8, 10 and 9, 12, 15.Masons use 3, 4, 5 with a string to set a square corner.

⚠️ Spot the hidden right angle

Exam figures don't always mark the 90°. In a 2017 O/L question many students missed that the angle in a semicircle is a right angle, so they never used Pythagoras. Look for right angles in rectangles, squares, walls on level ground, due north and due east, and angles in a semicircle. Sketch the triangle and mark them.

📌 Remember

• In a right triangle: hypotenuse² = sum of the squares of the other two sides.• Hypotenuse: opposite the right angle, longest side.• Finding the hypotenuse: add the squares. Finding another side: subtract. Then take the square root.• If c² = a² + b², the triangle is right-angled (3, 4, 5; 5, 12, 13; 8, 15, 17).• Look for hidden right angles and sketch first.

🧾 Quick revision sheet

Pythagoras' relationship	c² = a² + b² (c = hypotenuse)
What is the hypotenuse?	The side opposite the right angle; the longest side.
Finding a shorter side	a² = c² − b², then square root
Name three Pythagorean triples	3, 4, 5; 5, 12, 13; 8, 15, 17
Angle in a semicircle	Always 90°
Is 7, 24, 25 a right triangle?	Yes: 49 + 576 = 625 = 25²

Now test yourself: 5 quick checks and 2 more worked examples

The full lesson checks your answers as you go, and Gedara Sir tracks which parts you've got.

📖 Start the 10-minute lesson free ▶ Practise 5 past-paper questions