Restrictions on X

Question

XZ=12 , ZY=18 , XY= 4x-6

What are the restrictions on X?

Answer 1

To see if these figures form a triangle a^2+b^2=c^2 is the formula/rule for all triangles.

If a=12, b=18, c=4x-6 then 12^2+18^2=(4x-6)^2

144+324=16x^2-48x+36

432=16x^2-48x

x=6.9083

c=21.633

Answer 2

I'm just going to draw some random triangle based on the information provided:

So if this triangle were to be squashed flat, obviously it wouldn't be a triangle anymore but here is a version that's close to flat:

Can you see that the length of XY is getting very close to 12 + 18 (or 30) units long? The flatter it gets (the closer angle z gets to 180 degrees), the closer XY gets to 30. But, of course it can never equal 30 because then it wouldn't be a triangle any more (it would be a straight line). So we can say XY is < 30 or 4x - 6 < 30. Solving, we get:

4x - 6 < 30

4x < 36

x < 9

The same holds if I pull the Z vertex up, the triangle becomes like this:

As angle Z gets smaller and smaller, side XY gets closer and closer to 18 - 12 (or 6) units long. But it could never actually be 6 because then it wouldn't be a triangle any longer (it would be a straight line). So we can say XY > 6 or 4x - 6 > 6. Solving, we get:

4x - 6 > 6

4x > 12

x > 3

So the restrictions on x must be x > 3 and x < 9. We can combine these together and say 3 < x < 9 or we can just say it in English as: x must be larger than 3 but smaller than 9.