Answer:
11
Step-by-step explanation:
First of all, for what values of x can we have a triangle.
By triangle inequality, we have that 10-5<x<10+5.
Simplifying this gives us 5<x<15.
So the answer is either 6 or 11.
An acute triangle with sides
where
is the largest then
.
Now if the triangle is acute (with the assumption x is the largest) then:
![x^2](https://tex.z-dn.net/?f=x%5E2%3C5%5E2%2B10%5E2)
![x^2](https://tex.z-dn.net/?f=x%5E2%3C25%2B100)
![x^2](https://tex.z-dn.net/?f=x%5E2%3C125)
This implies that
with the condition that x>10 since we assumed it largest so the actual restriction on x is:
(
)
So this includes 11 and not 6.
Now if the triangle is acute (with the assumption x is not the largest) then:
![10^2](https://tex.z-dn.net/?f=10%5E2%3C5%5E2%2Bx%5E2)
![100](https://tex.z-dn.net/?f=100%3C25%2Bx%5E2)
![75](https://tex.z-dn.net/?f=75%3Cx%5E2)
![x^2>75](https://tex.z-dn.net/?f=x%5E2%3E75)
This means that
with condition x is less than 10 since we are assuming x is not the largest.
(
)
So this mean that x would have to be included between
and 10.
Either way 6 is not included in either of the acute triangle cases.
11 is the only one that satisfies the condition in at least one of the cases.
![11^2](https://tex.z-dn.net/?f=11%5E2%3C5%5E2%2B10%5E2)
![121](https://tex.z-dn.net/?f=121%3C25%2B100)
is true and 11 is a number between 5 and 15.