help me in all these for 50 POINTS

Hey there!

To start, the figure shown in the image is a square due to the fact that all of its sides are equivalent and the angles formed in the corners are all right angles.

Now, notice that there is a line running along the diagonal of the square with the length of 10. This line splits the square into two right triangles. 

Because you want to find the value of x, you must use the Pythagorean theorem: a^2+b^2=c^2
(a and b are the values of the legs and c is the value of the hypotenuse, in other words, the diagonal)

Now, plug in your values:
a=x
b=x
c=10

x^2+x^2=(10)^2

Now, simplify and combine your like terms (terms of the same variables and are raised to the same power):
x^2+x^2=10^2
2x^2=100

Now, solve for x:
2x^2=100 (divide both sides by two to isolate x^2)
÷ 2        ÷2
=
x^2=50   (now, square root both sides to isolate x)

[tex] \sqrt{x^2} [/tex]=[tex] \sqrt{50} [/tex]

This will result in x=[tex] \sqrt{50} [/tex].

However, you must simplify this square root in order to get your final answer. 

To simplify this root, you must factor [tex] \sqrt{50} [/tex] into the largest perfect square that is also a factor of the radical. This would be [tex] \sqrt{25} [/tex] * [tex] \sqrt{2} [/tex]. 

This should look like this: [tex] \sqrt{25} [/tex] * [tex] \sqrt{2} [/tex]

Now, simplify your perfect square to get 5[tex] \sqrt{2} [/tex] as your final answer.

Therefore, your final answer would be choice C.

Hope this helps and have a marvelous day! :)