Geometry Data Sufficency

[veronica1]

I don't understand how the assumption is made for statement one is made on a right triangle. That how does knowing one side is 9 yield a range of 31 to 49 for the other side. The square root of 1519 of 38.97... 1519 = (40)sq + (9)sq


In triangle XYZ, the length of side XY is 40. Is triangle XYZ a right triangle?

(1) The length of side YZ is 9.

(2) The length of side XZ is 41.

Statement (1): We are told that the length of side YZ is 9. This sets the range for possible lengths of the third side: 31 < XZ < 49. This range includes two configurations that would result in triangle XYZ being a right triangle but many more that would not.

[frankiesteinster]

Its hard to understand here how option 1 is enough to prove that it is a right angle triangle. Does the question has a diagram or some other information associated?

If this is the question, then answer should be C because sq(9) + sq(40) = sq (41).

Just to explain you what they mean in explanation " This sets the range for possible lengths of the third side: 31 < XZ < 49. "
"Sum of 2 sides of a triangle is always greater than the third"

which means XZ < (40+9) and greater than a value such that XZ+9 >40 => XZ>31
So, 31<XZ<49

[veronica1]

How do we come up with the number 31 and 49?

Its hard to understand here how option 1 is enough to prove that it is a right angle triangle. Does the question has a diagram or some other information associated?

If this is the question, then answer should be C because sq(9) + sq(40) = sq (41).

Just to explain you what they mean in explanation " This sets the range for possible lengths of the third side: 31 < XZ < 49. "
"Sum of 2 sides of a triangle is always greater than the third"

which means XZ < (40+9) and greater than a value such that XZ+9 >40 => XZ>31
So, 31<XZ<49

[atharshiraz]

How do we come up with the number 31 and 49?

Just to explain you what they mean in explanation " This sets the range for possible lengths of the third side: 31 < XZ < 49. "
"Sum of 2 sides of a triangle is always greater than the third"

which means XZ < (40+9) and ...

As "frank" mentioned.

If x,y,z are the vertices of a triangle and we know sides XY and YZ then side XZ is simply :

|XY - YZ |< XZ < (XY + YZ)

This theorem is given in the "geometry" guide.

So if you know two sides (40 and 9) then you can come up with a range for the third side XZ :
|40-9|< XZ < (40+9)
31 < XZ < 49

That is how you get the range 31 to 49.

[krishnan.anju1987]

Consider it this way, one side is x and the other sides as mentioned in the statement are 40 and 9. Now triangle inequality states that the sun of two sides must be greater than the third side.

This implies

40+9>x i.e 49>x
9+x>40 i.e x>31
40+x>9 which will always be true for any value of x unless x is negative which is not possible..


Thus, if three sides x,y,z exist and z is unknown and x and y are known, then |x-y| < z < |x+y|

[jnelson0612]

Hi all,
Per forum rules, we need an original source of this question to be able to keep this thread on the board. Thank you!