Final answer:
To find the area of an isosceles right triangle with a hypotenuse of x√6, we use the Pythagorean theorem to find the length of the legs and then apply the area formula for a triangle, resulting in an area of (3/2) × x² square units.
Step-by-step explanation:
Given that we have an isosceles right triangle with a hypotenuse of x√6 units, we can use the Pythagorean theorem to find the lengths of the other two equal sides. For an isosceles right triangle, the legs are congruent, and the Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c²) is equal to the sum of the squares of the other two sides (a² + b²).
Let both legs of the triangle have length 'a'. Thus, using the Pythagorean theorem:
a² + a² = (x√6)²
2a² = x² × 6
To find a single leg 'a', we can take the square root of both sides:
a = x√3
Now, knowing a single leg allows us to calculate the area of the triangle. The area formula for a triangle is 1/2 × base x height, which for our isosceles right triangle translates to 1/2 × a × a (since the base and height are the same length in this case).
Area = 1/2 × (x√3) × (x√3)
Area = 1/2 × x²×3
Therefore, the area of the triangle is (3/2) × x² square units.