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## Homework Statement

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Use vectors and the dot product to prove that the midpoint of the hypotenuse of a right triangle is equidistant to all three vertices.

## Homework Equations

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I know the dot product is

**A⋅B**= |

**A**||

**B**|cosΘ ...... or .... A1B1 + A2B2 + A3B3 ... + AnBn

I know the magnitude of a vector is its length, and is given by√(x^2 + y^2) I have omitted the z^2 since I want to solve this in R^2.

## The Attempt at a Solution

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I figure doing this in R^2 is easier, so I drew a right triangle with points P(0,0), R(0, y), S(x, 0)

I know the midpoint would be given by T = (x/2, y/2)

I constructed vectors using these points:

**PS**= <x, 0>

**PR**= <0, y>

**TS**= <x/2, -y/2>

**RT**= <x/2, -y/2>

**PT**= <x/2, y/2>

From this I can show that the magnitude of

**TS, RT**, and

**PT**are all the same, thus their distances are the same. I can also use this to show that

**TS**=

**RT**. I can then take the dot product of

**TS**and

**RT**and work it down to show the angle between them is 0 (so cosΘ = 1) and thus they are the same vector.

What I cannot do for the life of me is use the dot product to prove that

**PT**has the same magnitude as the other two vectors,

**TS**and

**RT**. I just can't figure it out. I've tried finding ways to express

**PT**as other vectors but everything I've tried has failed. I am tearing my hair out!

Here is a diagram of how I am trying to layout my triangle:

I would greatly appreciate any insight into this very troubling problem. :(

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