Prove √(a^2+b^2)≤ √[(x−a)^2+(y−b)^2]+√(x^2+y^2)

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Discussion Overview

The discussion centers around proving the inequality $$\sqrt{a^2+b^2}\leq\sqrt{(x-a)^2+(y-b)^2}+\sqrt{x^2+y^2}$$. Participants are exploring the mathematical validity of this statement, seeking both algebraic and visual representations.

Discussion Character

  • Mathematical reasoning, Debate/contested

Main Points Raised

  • Some participants request a visual representation to aid in understanding the inequality.
  • Others emphasize the need for an algebraic solution, indicating a preference for a purely mathematical approach.

Areas of Agreement / Disagreement

There is no consensus on the approach to take, as some participants are focused on visual aids while others insist on an algebraic proof.

Contextual Notes

Participants have not yet provided specific assumptions or definitions that may affect the proof, and the discussion remains open to various methods of proof.

solakis1
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prove:

$$\sqrt{a^2+b^2}\leq\sqrt{(x-a)^2+(y-b)^2}+\sqrt{x^2+y^2}$$
 
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solakis said:
prove:

$$\sqrt{a^2+b^2}\leq\sqrt{(x-a)^2+(y-b)^2}+\sqrt{x^2+y^2}$$

This comes from law of triangle sides

distance from origin to (a,b) < distance from (x,y) to (a,b) + distance from origin to (x,y) and equal if (x,y) is between (0,0) and (a,b) and in the line between the 2
 
kaliprasad said:
This comes from law of triangle sides

distance from origin to (a,b) < distance from (x,y) to (a,b) + distance from origin to (x,y) and equal if (x,y) is between (0,0) and (a,b) and in the line between the 2

Can you draw a picture ??

I was meaning an algebraic solution in my OP
 
solakis said:
Can you draw a picture ??

I was meaning an algebraic solution in my OP

we have $\sqrt{(x-a)^2+(y-b)^2} + \sqrt{x^2+y^2} = |(x-a) + i(y-b)| + | x + iy| = | (a-x) + (b-y) i | + | x + iy|$
$\ge |(a - x + x) + i(b-y+y)| = | a + ib| = \sqrt{a^2+b^2}$
 
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