if the altitude of an isosceles right triangle has a length of x units, what is the length of one leg of the…

Explanation:

Step1: Recall properties of isosceles right - triangle

In an isosceles right - triangle, the altitude drawn to the hypotenuse divides the isosceles right - triangle into two smaller congruent isosceles right - triangles. Let the large isosceles right - triangle have legs of length $a$ and hypotenuse $c$. The altitude $h$ to the hypotenuse of an isosceles right - triangle satisfies the following relationships.

Step2: Use the geometric mean theorem or properties of 45 - 45 - 90 triangles

In a 45 - 45 - 90 triangle, the ratio of the sides is $1:1:\sqrt{2}$. If the altitude of the isosceles right - triangle is $x$, then the two smaller isosceles right - triangles formed by the altitude have legs of length $x$. The leg of the large isosceles right - triangle is the hypotenuse of one of the smaller isosceles right - triangles. Using the Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}$ for a right - triangle. In a 45 - 45 - 90 triangle with legs of length $x$, if $a = x$ and $b = x$, then the hypotenuse (which is the leg of the large isosceles right - triangle) $l=\sqrt{x^{2}+x^{2}}=\sqrt{2x^{2}}=x\sqrt{2}$.

Answer:

B. $x\sqrt{2}$ units