Geometry and proof · Ages 13–14
Pythagoras' theorem
From “Finding the hypotenuse” to “Pythagorean triples”, step by step, each skill with its methods, worked examples and answers.
Skills and methods in this topic, easy to hard
- Finding the hypotenuseThe hypotenuse faces the right angle; c2 = a2 + b2, then the root
Method62 + 82 = 100, 100 = 10
Another way: A multiple of a triple6, 8 = 2 × (3, 4), so the hypotenuse = 2 × 5 = 10
- Finding a shorter sideA leg is a subtraction: b2 = c2 − a2
Method132 − 52 = 144, 144 = 12
Another way: Difference of squares132 − 52 = (13 + 5)(13 − 5) = 18 × 8 = 144
- The converse of Pythagoras' theoremShort sides' squares add to the longest's square ⇒ a right angle opposite it
Method92 + 122 = 225 = 152
Another way: Simplify first9, 12, 15 ÷ 3 = 3, 4, 5
- Using Pythagoras' theoremFind the right triangle and its hypotenuse, then add or subtract
MethodLadder 13 m, foot 5 m out: height = 132 − 52 = 12 m
Another way: Spot a hidden triple5, ?, 13 → 5, 12, 13
- Pythagorean triplesm2 − n2, 2mn, m2 + n2; or odd n, (n2 − 1) ÷ 2, (n2 + 1) ÷ 2
Methodm = 3, n = 2: 5, 12, 13
Another way: The odd-number rulen = 7: 7, 24, 25
Worked examples
From different skills in this topic. Have a go first, then open each one for the answer and the working.
Example 1Finding the hypotenuse →
Triangle ABC in the picture is right-angled. Which side is the hypotenuse?
- A. AB
- B. CA
- C. It has none
- D. BC
Choose the right answer
Show the answer and working
Answer: D. BC
Find the right angle (the one with the little square); the side facing it, not touching that corner, is the hypotenuse, the longest side.
Example 2The converse of Pythagoras' theorem →
A triangle with sides of 36 cm, 15 cm and 39 cm is right-angled. How long is the side opposite the right angle, in cm?
Show the answer and working
Answer: 39 cm
The right angle faces the longest side: 362 + 152 = 1521 = 392, so it is 39 cm.
Example 3Pythagorean triples →
7, 24, 25 is a Pythagorean triple, and so is 9 times it: , , .
Fill in each blank
Show the answer and working
Answer: 63, 216, 225
Times each by 9: (9 × 7)2 + (9 × 24)2 = 92 × (72 + 242) = 92 × 252 = (9 × 25)2.
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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