We have the following right triangle:
And we need to find the measure of the side represented by x.
To find it, we can proceed as follows:
1. Apply the Pythagorean Theorem as follows:
[tex]x^2+(5in)^2=(10in)^2[/tex]
We have that the sum of the squares of the legs (5 in and x inches) of the right triangle is equal to the square of the hypotenuse (10 in).
2. Then subtract (5in)^2 from both sides of the equation:
[tex]\begin{gathered} x^2+(5in)^2-(5in)^2=(10in)^2-(5in)^2 \\ \\ x^2=(10\text{in})^2-(5\text{in})^2 \end{gathered}[/tex]
3. Now, we can extract the square root to both sides of the equation (without using units):
[tex]\begin{gathered} \sqrt{x^2}=\sqrt{100-25}=\sqrt{75} \\ \\ x^=\sqrt{75} \end{gathered}[/tex]
4. Finally, to simplify the number at the right of the equation, we can factor the radicand as follows:
And these are the prime factors for 75. Then we can rewrite the value as follows:
[tex]\begin{gathered} 75=3*5^2 \\ \\ x=\sqrt{75}=\sqrt{3*5^2}=5\sqrt{3} \\ \\ x=5\sqrt{3} \end{gathered}[/tex]
Therefore, in summary, the value for x is:
[tex]x=5\sqrt{3}\text{ in}[/tex]
[Option B.]
