To determine the volume of a right triangular prism where the height is equal to the leg length of the base, we need to consider both the base and the height of the prism. Here is the step-by-step solution:
1. Identify the shape and dimensions of the base:
- The base of the prism is a right-angled triangle.
- Let [tex]\( x \)[/tex] be the leg length of the right-angled triangular base.
2. Calculate the area of the triangular base:
- The area [tex]\( A \)[/tex] of a right-angled triangle with legs of length [tex]\( x \)[/tex] is given by:
[tex]\[
A = \frac{1}{2} \times (\text{leg}_1) \times (\text{leg}_2)
\][/tex]
Since both legs are equal to [tex]\( x \)[/tex], the equation becomes:
[tex]\[
A = \frac{1}{2} \times x \times x = \frac{1}{2} x^2
\][/tex]
3. Identify the height of the prism:
- According to the problem statement, the height [tex]\( h \)[/tex] of the prism is equal to the leg length of the triangular base, which is [tex]\( x \)[/tex].
4. Compute the volume of the prism:
- The volume [tex]\( V \)[/tex] of a prism is given by the product of the base area [tex]\( A \)[/tex] and the height [tex]\( h \)[/tex]:
[tex]\[
V = A \times h
\][/tex]
- Substituting the area and the height we previously found:
[tex]\[
V = \left(\frac{1}{2} x^2\right) \times x = \frac{1}{2} x^3
\][/tex]
Therefore, the expression that represents the volume of the prism in cubic units is:
[tex]\[
\frac{1}{2} x^3
\][/tex]
So, the correct answer is:
[tex]\[
\frac{1}{2} x^3
\][/tex]
