To check if it is a right triangle, apply the Pythagorean theorem: - Baxtercollege

Understanding the Context

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, known as the legs.
Mathematically, it’s written as:

> a² + b² = c²

Where:

• a and b are the lengths of the legs,
  
• c is the length of the hypotenuse.

Step-by-Step Guide to Using the Pythagorean Theorem

Key Insights

Checking if a triangle is right-angled using this theorem involves three straightforward steps:

1. Identify the longest side
   Begin by measuring or determining the longest side of the triangle, as this will always be the hypotenuse if the triangle is right-angled.

2. Apply the Pythagorean equation
   Square the lengths of both legs and add them:
   

   • Calculate a²
     
   • Calculate b²
     
   • Sum a² + b²

Do the same for the hypotenuse (if known): square its length and write c².

3. Compare both sides
   If a² + b² equals c², then the triangle is a right triangle.
   If they are not equal, the triangle does not have a right angle and is not right-angled.

Final Thoughts

Practical Example

Consider a triangle with sides 3 cm, 4 cm, and 5 cm.

• The longest side is 5 cm → assumed hypotenuse (c).
  
• Calculate:
  a² + b² = 3² + 4² = 9 + 16 = 25
  c² = 5² = 25

Since a² + b² = c², this is a right triangle. You’ll recognize this classic 3-4-5 Pythagorean triple!

When Is the Theorem Not Enough?

The Pythagorean Theorem only applies to right triangles. For triangles that aren’t right-angled, other methods like the cosine rule are required. But if you suspect a triangle might be right-angled, squaring and comparing sides remains the quickest and most definitive check.