To solve for the length of one leg of the large right triangle in terms of [tex]\( x \)[/tex], we need to understand the properties of an isosceles right triangle.
An isosceles right triangle has two sides of equal length, which we'll call [tex]\( a \)[/tex], and the right angle formed between them. The hypotenuse is opposite the right angle and can be found using the Pythagorean theorem.
### Step-by-Step Solution:
1. Identify the Altitude and the Hypotenuse:
- The given isosceles right triangle has an altitude of length [tex]\( x \)[/tex] units.
- In a larger right triangle formed by this altitude (presuming the altitude divides the isosceles right triangle into two smaller right triangles), the lengths of both legs would be equal to the hypotenuse of smaller right triangles.
2. Express in Terms of [tex]\( x \)[/tex]:
- For the half part of the isosceles triangle, each leg of the smaller right triangle is [tex]\( \frac{x}{\sqrt{2}} \)[/tex]. Because the altitude directly comes from the hypotenuse, however, it depicts equal proportions applied on all edges being scaled up solutions simplifies to a factor applied homogeneously.
3. Calculate the Hypotenuse of Smaller Right Triangle:
- Recognizing every leg mirrors to the basic Pythagorean form as [tex]\(a^2 + a^2 = (x)^2 \rightarrow 2a^2 = x^2 \rightarrow a = \frac{x}{\sqrt{2}} \)[/tex].
### Final Conclusion:
Thus, for an isosceles right triangle, if the length of one leg is [tex]\( x \)[/tex], the hypotenuse becomes [tex]\( \sqrt{2} \cdot x \)[/tex].
Hence, the length of one leg of the large right triangle, simplifies to [tex]\( x \sqrt{2} \)[/tex].
Therefore, the correct answer is:
[tex]\[ x \sqrt{2} \text{ units} \][/tex]
