MathsTriangles Inequality Theorem

Triangles Inequality Theorem

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    • Triangle Inequality Theorem
    • Application of Triangle Inequality Theorem
    • What’s Next?

    In the previous segment, we solved an example based on the Angle sum property of a triangle. In this segment, we will use the Triangle inequality theorem to prove an interesting property of a triangle.

    What is the Triangle inequality theorem?

    The triangle inequality theorem states that ‘the sum of the lengths of any two sides of a

    triangle is greater than the third side’.

    This means for any triangle ABC,

    • AB + BC > CA
    • BC + CA > AB
    • AB + CA > BC

    Application of triangle inequality theorem

    Let us now use the triangle inequality theorem to prove that the sum of lengths of the sides of a triangle is greater than twice the length of the segment joining a vertex to the opposite side.

    That is, in △ABC, to prove that AB + BC + CA > 2AD.

    Figure for Proof

    Statement Reason
    1 In △ABD,

    • AB + BD > AD
    • AD + BD > AB
    • AB + AD > BD
    Triangle inequality theorem
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