The given triangle is an isosceles right triangle, meaning that the two legs are equal in length. The hypotenuse is labeled as 6, and the unknown leg is x.

Step 1: Use the Pythagorean Theorem

For a right triangle, the Pythagorean theorem states:

$a^2 + b^2 = c^2$

Since the triangle is isosceles, the two legs are equal, so let $a = b = x$ and $c = 6$:

$x^2 + x^2 = 6^2$

$2x^2 = 36$

Step 2: Solve for $x$

$x^2 = \frac{36}{2} = 18$

$x = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

Approximating the value:

$x \approx 3 \times 1.414 = 4.2$

Final Answer:

$\mathbf{x \approx 4.2}$

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Here are five related questions you might find helpful:

1. How do you determine if a triangle is isosceles?
2. What is the Pythagorean theorem, and how is it used?
3. How do you find the hypotenuse of an isosceles right triangle?
4. What are the properties of a 45-45-90 triangle?
5. How do you simplify square roots in calculations?

Tip: In an isosceles right triangle, the legs are always in the ratio $1:1:\sqrt{2}$, making calculations easier! 😊