Question

prove that the sum of the lengths of the diagonals is less than the perimeter of the quadrilateral.​

Answer 1

Answer:

Take the longer of the two diagonals.

If you have two sides of a triangle, the third side must be shorter than the sum of the other two sides.

That diagonal forms the third side of two triangles, one with two sides of the quadrilateral, and one with the other two sides of the quadrilateral.

Let the sides be AB, and CD respectively, and x be the diagonal.

x <A+B

x <C+D

x+x <A+B+C+D

Lastly, remember that we chose the larger diagonal. Let y be the other diagonal.

y <x

y+x <x+x

y+x<A+B+C+D

QED