Scalene Triangle

Definition of scalene triangle: A scalene triangle is a type of triangle that has three unequal sides and three unequal angles. In other words, all the sides and angles of a scalene triangle have different measures. The term "scalene" comes from the Greek word "skalenos", meaning "uneven" or "unequal".

In the figure above, all the three sides and angles of △PQR are unequal, hence it is scalene triangle.

Characteristics of a Scalene Triangle:

• Side Lengths: In a scalene triangle, each side has a different length. Let's label the sides as a, b, and c. So, a ≠ b ≠ c.
• Angle Measures: The angles of a scalene triangle are also different from one another. Let's label the angles as ∠A, ∠B, and ∠C. So, ∠A ≠ ∠B ≠ ∠C.
• Triangle Inequality: The triangle inequality theorem applies to scalene triangles, just as it does for all triangles. It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In mathematical terms, for a scalene triangle with sides a, b, and c: a + b > c, b + c > a, and c + a > b.
• Angle Relationships: Since the angles of a scalene triangle are all different, there are no special relationships between the angles. Each angle have different measure.
• Perimeter: The perimeter of a scalene triangle is simply the sum of the lengths of its three sides, i.e., a + b + c.
• Area: The area of a scalene triangle can be calculated using various methods, such as Heron's formula or by calculating the height and base. Heron's formula states that the area (A) of a scalene triangle with side lengths a, b, and c is given by:
  $A=\sqrt{s(s-a)(s-b)(s-c)}$
  where s is the semi-perimeter of the triangle, calculated as$s={\displaystyle \frac{(a+b+c)}{2}}$.
• Congruence: Scalene triangles are not congruent to each other unless they have the same side lengths and angle measures. Congruent triangles have exactly the same shape and size.
• Symmetry: Scalene triangles do not possess any line or rotational symmetry. They can be rotated and reflected in various ways.