In mathematics, an inequality is an expression that indicates that one quantity is greater or less than another. A geometric inequality it is an inequality that involves different geometric measures such as angles, areas, lengths, among others. There are different types of geometric inequalities based on different geometric shapes.
In any triangle the sum of the lengths of any two sides is always greater than the length of the remaining third side. Suppose that a, b and c are the lengths of the three sides of a triangle, therefore:
- (a + b)> c
- (b + c)> a
- (a + c)> b
Example: In a triangle let a = 8 cm and b = 3 cm, determine the possible length of the third side c.
We know that:
- (a + b)> c → (8 + 3)> c → 11> c
- (b + c)> a → (3 + c)> 8 → c> 5
- (a + c)> b → (8 + c)> 3 → c> -5
We ignore this last result (3) because the lengths must always be positive, therefore:
3 <c <11
The third side is greater than 3 cm or less than 11 cm.
The Pythagorean theorem states that in a right triangle with sides of length a ≤ b ≤ c, a2+ b2 = c2. The Pythagorean inequality It is a generalization of the Pythagorean Theorem, which extends to obtuse and acute triangles:
- Acute triangle: In an acute triangle, the square on the longest side is always less than the sum of the squares on the other two sides. That is, in an acute triangle with sides of length a ≤ b ≤c, a2+ b2 > c2.
- Obtuse triangle: In an obtuse triangle, the square on the longest side is always greater than the sum of the squares on the other two sides. That is, in an obtuse triangle with sides of length a ≤ b ≤ c, a2 + b2 <c2.
Trigonometric inequalities can be written as:
- [f, g, h, ...]> 0
- [f, g, h, ...] <0
- [f, g, h, ...] ≤ 0
- [f, g, h, ...] ≥ 0
Where f, g, h, ... denote trigonometric functions.
We can solve for the variable, that is, the values of x that make the inequality true; these values will be the solutions of the given inequality. For example:
2tanθ + tan 2θ> 3cot θ