Proof of Triangle Inequality: a+b-c, b+c-a, c+a-b

In summary, the given inequality states that for any triangle with side lengths a, b, and c, the sum of the square roots of the differences of each pair of sides is always less than or equal to the sum of the square roots of the original side lengths. This can be proven by showing that each individual term in the inequality is less than or equal to twice the respective side length.
  • #1
Albert1
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Let a, b, c be the lengths of the sides of a triangle. Prove that:
$\sqrt{a+b-c}$+$\sqrt{b+c-a}$+$\sqrt{c+a-b}\leq\sqrt{a}+\sqrt{b}+\sqrt{c}$
 
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  • #2
Albert said:
Let a, b, c be the lengths of the sides of a triangle. Prove that:
$\sqrt{a+b-c}$+$\sqrt{b+c-a}$+$\sqrt{c+a-b}\leq\sqrt{a}+\sqrt{b}+\sqrt{c}$

$(\sqrt{a+b-c}$+$\sqrt{b+c-a})^2\leq 4b$
$\therefore \sqrt{a+b-c}$+$\sqrt{b+c-a}\leq 2\sqrt{b}--------------(1)$
likewise
$\therefore \sqrt{b+c-a}$+$\sqrt{c+a-b}\leq 2\sqrt{c}--------------(2)$
$\therefore \sqrt{a+b-c}$+$\sqrt{c+a-b}\leq 2\sqrt{a}--------------(3)$
(1)+(2)+(3) the proof is done
 

1. What is the proof of the triangle inequality?

The proof of the triangle inequality states that in a triangle, the sum of any two sides is always greater than the third side.

2. Why is the triangle inequality important in mathematics?

The triangle inequality is important in mathematics because it is a fundamental principle used in many different branches of mathematics, such as geometry, trigonometry, and calculus. It also has practical applications in fields like engineering and physics.

3. How can the triangle inequality be visualized?

The triangle inequality can be visualized by drawing a triangle and measuring the lengths of each side. It can also be represented geometrically by using points and lines to show the relationship between the sides of the triangle.

4. Is the triangle inequality always true?

Yes, the triangle inequality is always true. It is a fundamental property of triangles in Euclidean geometry and cannot be disproven.

5. Can the triangle inequality be applied to any type of triangle?

Yes, the triangle inequality can be applied to any type of triangle, including equilateral, isosceles, and scalene triangles. It is a universal principle that applies to all triangles regardless of their size or shape.

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