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A triangle has side lengths of 1, 1 - c, and 1 - 2c.  What are all the possible values of c?

 May 14, 2020
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So a triangle's legs' sum must exceed the hypothenuse to be a triangle.

 

First, assuming c is positive, we have 1-2c and 1-c as the legs as they are each smaller than 1.

 

So,.

 

1-2c + 1-c >1

2-3c >1

1 >3c

1/3 >c

c < 1/3

 

That's after a bit of rearrangement.

 

NOW assuming c is 0. Hmm. That doesn't work since 1+1 = 2. Hmm.

 

NOW assuming that c is negative.

 

1-2c must be the hypothenuse as it is the largest, and 1 and 1-c are the legs.

 

1 + 1-c > 1-2c

2-c > 1-2c

1 > 2c

1/2 > c

 

c < 1/2

 

since c is smaller than 1/2 AND 1/3, we go with 1/3 since it's smaller than and the smallest.

 

c <1/3

 

If you don't understand anythign feel free to ask.

 May 14, 2020

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