32. The Scalene Triangle · three sides, three different lengths

Try to break it

Try dragging the corners until two sides look the same. What happens to the angles?

How you build it

Make a scalene triangle.

• Place point A.
• Place point B at a different distance from A.
• Place point C such that AC ≠ AB and BC ≠ AB ≠ AC. All three sides will end up different.
• Draw side AB.
• Draw side BC.
• Draw side CA. Triangle ABC is scalene — three different side lengths, three different angles.

The proof, step by step

Prove that a scalene triangle has no line of symmetry.

• By definition, a scalene triangle has all three sides of different lengths.
• A line of symmetry folds a shape onto itself, so it must match each side exactly onto another side of the same length.
• But a scalene triangle has no two sides of equal length — no side has a partner to fold onto.
• With no matching pair of sides to fold together, a scalene triangle can have no line of symmetry.

Worked example

In a triangle ABC, AB = 5 cm, BC = 7 cm, and CA = 6 cm. What type of triangle is ABC?

Since all three sides (5 cm, 7 cm, and 6 cm) have different lengths, triangle ABC is a scalene triangle.

• Isosceles triangle
• Equilateral triangle
• Scalene triangle — correct
• Right-angled triangle