Understanding Obtuse Triangles
An obtuse triangle is a triangle in which one interior angle measures greater than 90°. The other two angles must each be acute (less than 90°). A triangle can have at most one obtuse angle.
Identifying an Obtuse Triangle from Sides
Given three sides a, b, and c (where c is the longest side), a triangle is obtuse if and only if:
c² > a² + b²
When c² = a² + b², the triangle is right. When c² < a² + b², all angles are acute.
Area Calculation: Heron's Formula
s = (a + b + c) / 2
Area = √[s · (s - a) · (s - b) · (s - c)]
Area = √[s · (s - a) · (s - b) · (s - c)]
Finding Angles: Law of Cosines
A = arccos[(b² + c² - a²) / (2bc)]
B = arccos[(a² + c² - b²) / (2ac)]
C = arccos[(a² + b² - c²) / (2ab)]
B = arccos[(a² + c² - b²) / (2ac)]
C = arccos[(a² + b² - c²) / (2ab)]
Properties of Obtuse Triangles
- Altitude Position: In an obtuse triangle, two altitudes fall outside the triangle.
- Circumcenter: The circumcenter lies outside the triangle.
- Orthocenter: The orthocenter lies outside the triangle.
- Longest Side: The side opposite the obtuse angle is always the longest side.
Alternative Area Formulas
- Base-Height: Area = ½ × base × height. For obtuse triangles, the height may extend outside the triangle.
- SAS Method: Area = ½ · a · b · sin(C). Works for any included angle, including obtuse.
- Coordinate Method: Area = ½|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
Real-World Examples
- Architecture: Many roof designs use obtuse triangles for aesthetic appeal.
- Land Surveying: Irregularly shaped plots often form obtuse triangles.
- Engineering: Truss structures sometimes incorporate obtuse triangles to distribute loads.
- Navigation: Position fix calculations with obtuse triangle geometry.