To determine the length of one leg of an isosceles right triangle given its altitude, let's break down the problem step-by-step.
### Step-by-Step Solution:
1. Understanding the Isosceles Right Triangle:
An isosceles right triangle is a right triangle where the two legs are of equal length, and the angles are [tex]\(45^\circ\)[/tex], [tex]\(45^\circ\)[/tex], and [tex]\(90^\circ\)[/tex].
2. Isosceles Right Triangle Division:
When we draw the altitude from the right angle (90-degree angle) to the hypotenuse, we split the isosceles right triangle into two smaller 45-45-90 right triangles.
3. Properties of a 45-45-90 Triangle:
In a 45-45-90 triangle:
- The legs are of equal length.
- Each leg’s length is [tex]\( x \)[/tex].
- The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of each leg.
4. Altitude and Hypotenuse Relation:
The altitude we draw acts as one of the legs of these two smaller 45-45-90 triangles.
5. Finding the Length of One Leg of the Large Right Triangle:
Since the altitude [tex]\( x \)[/tex] serves as a leg for the smaller 45-45-90 triangles, it implies that:
- The original leg of the large triangle is equal to the hypotenuse of one of the smaller triangles.
6. Hypotenuse of the 45-45-90 Triangle:
The formula for the hypotenuse of a 45-45-90 triangle can be given as:
[tex]\[
\text{Hypotenuse} = \text{Leg} \times \sqrt{2}
\][/tex]
In our situation:
[tex]\[
\text{Leg length of the large triangle} = x \times \sqrt{2}
\][/tex]
After following these logical steps, we conclude that the length of one leg of the large isosceles right triangle, in terms of [tex]\( x \)[/tex], is:
[tex]\[
x \sqrt{2} \text{ units}
\][/tex]
Thus, the correct option is [tex]\( x \sqrt{2} \text{ units} \)[/tex].
