To find the length of one leg of an isosceles right triangle when given the altitude [tex]$x$[/tex], follow these steps:
1. Understand the Properties of an Isosceles Right Triangle:
In an isosceles right triangle, the two legs are congruent (the same length), and the angle between them is [tex]$90^\circ$[/tex]. The hypotenuse is opposite this right angle and is the longest side of the triangle.
2. Identify the Relationship Between the Legs and Hypotenuse:
Let's denote the length of each leg as [tex]$a$[/tex]. According to the Pythagorean Theorem, the length of the hypotenuse [tex]$h$[/tex] in an isosceles right triangle is:
[tex]\[
h = a\sqrt{2}
\][/tex]
3. Relation of Altitude to the Hypotenuse:
When an altitude is drawn to the hypotenuse of an isosceles right triangle, it forms two congruent right triangles, each with legs of [tex]$x$[/tex] units and hypotenuse of [tex]$\frac{a\sqrt{2}}{2}$[/tex] (since the altitude splits the hypotenuse equally).
4. Express the Legs in Terms of Altitude:
If the altitude is [tex]$x$[/tex], then the relationship can be set as:
[tex]\[
\left( \frac{a\sqrt{2}}{2} \right) = x
\][/tex]
5. Solve for the Length of the Leg [tex]\(a\)[/tex]:
[tex]\[
\frac{a\sqrt{2}}{2} = x
\][/tex]
Rearrange the equation to solve for [tex]$a$[/tex]:
[tex]\[
a\sqrt{2} = 2x
\][/tex]
[tex]\[
a = \frac{2x}{\sqrt{2}}
\][/tex]
Simplify [tex]\(\frac{2x}{\sqrt{2}}\)[/tex] by multiplying the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
a = \frac{2x \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}
\][/tex]
[tex]\[
a = \frac{2x \sqrt{2}}{2}
\][/tex]
6. Final Simplicity:
[tex]\[
a = x \sqrt{2}
\][/tex]
Therefore, the length of one leg of the isosceles right triangle in terms of [tex]$x$[/tex] is [tex]\( \boxed{x \sqrt{2}} \)[/tex] units.
