Don't make mistakes! #4

Geometry Level 4

Real number x x satisfies the condition below:

  • A triangle formed with side lengths of x + 4 x+4 , x + 5 x+5 and 2 x + 5 2x+5 is a right triangle.

How many values of x x are there?

This problem is a part of <Don't make mistakes!> series .

2 0 1 4 3

Boi (보이) by Boi (보이) , 19, South Korea

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1 solution

Marta Reece
Jun 22, 2017

Either x + 5 x+5 or 2 x + 5 2x+5 could be the hypotenuse, so there are two equations for x x

( x + 4 ) 2 + ( 2 x + 5 ) 2 = ( x + 5 ) 2 (x+4)^2+(2x+5)^2=(x+5)^2 with solutions x = 2.37 x=2.37 and x = − 3.37 x=-3.37

( x + 4 ) 2 + ( x + 5 ) 2 = ( 2 x + 5 ) 2 (x+4)^2+(x+5)^2=(2x+5)^2 with solutions x = − 1.22 x=-1.22 and x = − 3.28 x=-3.28

The second solutions in both cases are too large in absolute value for 2 x + 5 2x+5 to be a positive number, so only 2 \boxed2 solutions remain.

3 Helpful 0 Interesting 0 Brilliant 0 Confused

It would be interesting to find a case when there are 4 solutions.

Perhaps, we if had different expressions, we could even find 8 solutions!

Calvin Lin Staff - 3 years, 11 months ago

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@Calvin Lin Sir , what could have been a case with 8 solutions?

I think we can have a maximum of 6 solutions as either of the three expressions would be the hypotenuse and hence 3 equations which means maximum of 6 solutions..Am I right , sir?

Ankit Kumar Jain - 3 years, 11 months ago

@Marta Reece Although you got it right but I think you will have to verify that the condition of the either expression being a hypotenuse is satisfied by the solutions or not.

Ankit Kumar Jain - 3 years, 11 months ago

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