The sides of a triangle have lengths 4x+1, 2x+1, and 6x1. If the length of the longest side is 6x1, what values of x make the triangle obtuse?
meagan18
2011/01/07 00:58:09


7 votes


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11 votes


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I NEEED HELP
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Fun
2014/04/19 17:08:26
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The only way I could see how to gain the result is by using Pythagoras' principles of right angled triangles:
HAVE NOW WORKED OUT THE CORRECT ANSWER. USING 2 METHODS, I ARRIVED AT THE SAME  SO SHOULD BE CORRECT => X>1.54
[The 2 Methods: Completing the Square AND the Quadratic Formula]
If you begin by using Pythagoras formula [below] it will set you on the right track.
Pythagoras states that a^2 b^2 = c^2 in a rightangled Tri...
But really, we want the largest angle > than 90 deg, because obtuse ang's > than 90, by def'n.
To achieve this in our equation, the c^2 must be Greater than the SUM of the squares of the other 2 sides, not just EQUAL to them. In diagram, the 6x1 side has to be > than in the regular equation ... which would mathematically alter the Pythag formula thus:
(6x1)^2 > (4x 1)^2 (2x 1)^2
[Please Note: Ot is hard to explain this w/o Diagram, adn even my Mum always said I was a Diagram manLOL!...
Basically, we are beginning with a 90 deg tri  using Pythag ... but THEN, if we make the longest side even longer (meaning "greater than" ...
>
>
>>
>
>
>
>
The only way I could see how to gain the result is by using Pythagoras' principles of right angled triangles:
HAVE NOW WORKED OUT THE CORRECT ANSWER. USING 2 METHODS, I ARRIVED AT THE SAME  SO SHOULD BE CORRECT => X>1.54
[The 2 Methods: Completing the Square AND the Quadratic Formula]
If you begin by using Pythagoras formula [below] it will set you on the right track.
Pythagoras states that a^2 b^2 = c^2 in a rightangled Tri...
But really, we want the largest angle > than 90 deg, because obtuse ang's > than 90, by def'n.
To achieve this in our equation, the c^2 must be Greater than the SUM of the squares of the other 2 sides, not just EQUAL to them. In diagram, the 6x1 side has to be > than in the regular equation ... which would mathematically alter the Pythag formula thus:
(6x1)^2 > (4x 1)^2 (2x 1)^2
[Please Note: Ot is hard to explain this w/o Diagram, adn even my Mum always said I was a Diagram manLOL!...
Basically, we are beginning with a 90 deg tri  using Pythag ... but THEN, if we make the longest side even longer (meaning "greater than" in our equation), we will automatically open up the 90 deg angle to make it 'wider'  this will therefore make this angle "Obtuse"  which is the point of the exercise]
[You may figure it from here but I'll keep working. It's at this point, you may expand and simplify, THEN you may solve by one of the 2 methods mentioned]
Begin by expanding the 3 perfect squares => then it simplifies to
36x^212xPLUS1 > 20x^2 PLUS12x PLUS2
=> 16x^224x1 > 0 [a Quadratic equation]
So, using "Quadratic formula": x = [b PLUS/ sq.rt.(b^24ac)] / 2a
FOR SOME REASON, THE SH PROGRAM IS NOT INCLUDING THE "PLUS" SINGS i AM ENTERING ... SO I WILL HAVE TO WORD IN THE PLUS SIGNS  Sorry]
=> x = {24 PLUSsq.rt.(576 PLUS64)} / 32
=> x = 3/4 PLUSsq.rt.(5/8)
=> x = 1.54...approx
Therefore, x > 1.54...
The answer is now correctLOL
Perseverence, my Friend ;)
I worked it out at top.
Try Pythagoras
Sorry.
c^2 = a^2 + b^2  2abcosA , where A is an angle
so set that up with the the sides a,b,c as 2x+1 ect
then solve for the angle in terms of x
that formula can be rewritten
a^2 = b^2 + c^2  2bacosB,
b^2 = a^2 + c^2  2accosC,
and do the same thing, in each case, solving for the angle in terms of X
we know that A+B+C has to = 180, so plug in the values you got from the above formulas into that, and solve for x
I've got to prepare a Sunday School lesson for today [only partly done] ... so when I get a chance, I'll give it a go => Did u work it all out down to a value answer yourself??
Neither am ILOL!
We have no values to work with!HaHa!
Do you think the idea is workable at all??
I think you'd have to really work it carefully through in the mind, because you might be able to use an 'inequality' sign instead of an = somewhere.
ie
x=2
OK ... Ok ... Well, if u can't do it, hand it over to LB and that should make you jealousLOL!
Well sometimes we experince great Grief too :(