How to generate a right-angled triangle with integer sides (Pythagorean triple)

Use this method when you want to create geometry or trigonometry exercises involving right-angled triangles with integer side lengths.

Following this guide, you will generate a right-angled triangle whose sides form a (possibly scaled) Pythagorean triple, ensuring all side lengths are integers.

See it in action: Watch how this logic is implemented inside LearningLemur:

Before you begin

Requirements

• Basic knowledge of how to create a LearningLemur question
• Familiarity with adding an algorithm to generate random variables

Steps

Generate valid parameters $m > n$

n_val = random(1,5)
m_val = random(n_val + 1, n_val + 5)

This guarantees that $m > n$, a necessary condition for Euclid's formula.

Enforce coprimality and parity conditions.

while gcd(m_val, n_val) != 1 || (odd?(m_val)=true && odd?(n_val)=true) 
    n_val = random(1,5)
    m_val = random(n_val + 1, n_val + 5)
end

This ensures:

• $m$ and $n$ are coprime
• They are not both odd.

Together, these conditions generate a primitive Pythagorean triple.

Apply Euclid's formula.

a = m_val^2 - n_val^2
b = 2 * m_val * n_val
c = m_val^2 + n_val^2

This produces a right triangle with integer sides.

Optionally scale the triple.

k = random(1,3)
a = k*a
b = k*b
c = k*c

Scaling allows you to generate non-primitive triples as well.

Verify it worked

• Preview the question multiple times.
• Confirm that $a^2 + b^2 = c^2$.
• Ensure all sides are integers.
• Verify that no degenerate triangle appears.

Full algorithm (copy-paste version)

Use the complete version below if you want to copy the logic directly into your question algorithm:

# Generate two random integers m and n for Euclid's formula (m > n)
n_val = random(1,5)
m_val = random(n_val + 1, n_val + 5)

# Ensure m and n are coprime and not both odd
while gcd(m_val, n_val) != 1 || (odd?(m_val)=true && odd?(n_val)=true) 
    n_val = random(1,5)
    m_val = random(n_val + 1, n_val + 5)
end

# Use Euclid's formula to generate a Pythagorean triple
a = m_val^2 - n_val^2
b = 2 * m_val * n_val
c = m_val^2 + n_val^2

# Optionally, scale the triple by a random factor k
k = random(1,3)
a = k*a
b = k*b
c = k*c

Options and variations

• If you want only primitive triples, you can remove the scaling section (k).
• If you want larger triangles, you can increase the random range for m_val and n_val
• If you want to avoid very large numbers, you can reduce the upper bound of the interval.

Common errors

Generated sides are not integers.
Ensure Euclid's formula is written exactly as shown

Repeated regeneration in the while loop.
Broaden the interval for m_val and n_val

Unexpected degenerate triangles.
Confirm that the condition m_val > n_val is enforced

Related

• Understanding Advanced Logic in LearningLemur
• Common Patterns and Best Practices
• Glossary of Commands