Pythagorean triads
Identify Pythagorean triads, recognise triad families, and use triad knowledge to solve problems without a calculator.
Worked examples
Identifying a Pythagorean triad
Straightforward
Problem
Determine whether is a Pythagorean triad.
1
Square each number.
2
Check whether the sum of the two smaller squares equals the largest square.
3
State the conclusion.
Since , the numbers form a Pythagorean triad.
Answer
Yes, is a Pythagorean triad.
Finding the missing value in a triad
Moderate
Problem
Find the missing value in the Pythagorean triad .
1
Square the two known values.
2
Add the squares of the two legs.
3
Find the square root to get the hypotenuse.
Answer
The missing value is . The triad is .
Using triad knowledge to solve a problem
Challenging
Problem
A right-angled triangle has legs of m and m. Find the hypotenuse without using a calculator.
1
Recall common Pythagorean triads and check whether the given legs match one.
The triad is a known Pythagorean triple: .
2
Identify the hypotenuse from the triad.
Since , the hypotenuse is m.
Answer
The hypotenuse is m.
Practise
Q1·Straightforward
Do the numbers , , form a Pythagorean triad?
Explanation
, so yes, is a Pythagorean triad.
Q2·Straightforward
Do the numbers , , form a Pythagorean triad?
Explanation
, so yes, is a Pythagorean triad.
Q3·Straightforward
Do the numbers , , form a Pythagorean triad?
Explanation
, so yes, is a Pythagorean triad. It is the triple scaled by .
Q4·Straightforward
Do the numbers , , form a Pythagorean triad?
Explanation
, so is not a Pythagorean triad.
Q5·Moderate
The triad is scaled by a factor of to give . What is the missing value?
Explanation
Multiplying by : , , . The missing value is .
Q6·Moderate
Find the missing value in the Pythagorean triad .
Explanation
, so the missing value is .
Q7·Moderate
Find the missing value in the Pythagorean triad .
Explanation
, so the missing value is .
Q8·Moderate
The triad is scaled to give . What is the missing value?
Explanation
The scale factor is (since and ). So the hypotenuse is .
Q9·Challenging
A ladder m long leans against a wall with its foot m from the base. How high up the wall does the ladder reach? Give your answer without a calculator.
Explanation
This is the Pythagorean triad. The height up the wall is m.
Q10·Challenging
A rectangular field is m wide and m long. What is the length of the diagonal, in metres? Give your answer without a calculator.
Explanation
The diagonal is the hypotenuse of a right-angled triangle with legs and . Recognising the triad, the diagonal is m.
Q11·Challenging
A right-angled triangle has legs of cm and cm. What is the length of the hypotenuse, in centimetres? Give your answer without a calculator.
Explanation
is the triad scaled by . The hypotenuse is cm.
Q12·Challenging
A right-angled triangle has a hypotenuse of m and one leg of m. Find the length of the other leg, in metres. Give your answer without a calculator.
Explanation
is the triad scaled by . The other leg is m.