To determine the volume of a right triangular prism with the given conditions, follow these steps:
1. Identify the dimensions of the base triangle:
The base of the prism is a right triangle with both legs of length [tex]\( x \)[/tex], and the height of the prism is also [tex]\( x \)[/tex].
2. Calculate the area of the base triangle:
The area of a right triangle is given by:
[tex]\[
\text{Area of base} = \frac{1}{2} \times \text{base} \times \text{height}
\][/tex]
Since both the base and height of the triangle are [tex]\( x \)[/tex]:
[tex]\[
\text{Area of base} = \frac{1}{2} \times x \times x = \frac{1}{2} x^2
\][/tex]
3. Calculate the volume of the prism:
The volume of a prism is given by:
[tex]\[
\text{Volume} = \text{Base area} \times \text{height of the prism}
\][/tex]
Here, the base area is [tex]\( \frac{1}{2} x^2 \)[/tex] and the height of the prism is [tex]\( x \)[/tex]:
[tex]\[
\text{Volume} = \left(\frac{1}{2} x^2\right) \times x = \frac{1}{2} x^3
\][/tex]
Thus, the expression that represents the volume of the triangular prism is:
[tex]\[
\frac{1}{2} x^3
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{1}{2} x^3}
\][/tex]
