In a right triangle $ABC$ with $A$ as the right angle and the triangle is isosceles, what is the length of each leg if the hypotenuse is $16$ units?

• A. $40$ units
• B. $11$ units
• C. $23$ units
• D. $8\sqrt{2}$ units

Answer: (D)

Verified step by step guidance

1. Identify the properties of the triangle: Since triangle ABC is right-angled at A and isosceles, the two legs adjacent to the right angle are equal in length. Let's denote each leg as length \( x \).
2. Recall the Pythagorean theorem for right triangles: The square of the hypotenuse equals the sum of the squares of the legs. Mathematically, this is expressed as \( \text{hypotenuse}^2 = \text{leg}_1^2 + \text{leg}_2^2 \).
3. Substitute the known values into the Pythagorean theorem: Since both legs are equal, the equation becomes \( 16^2 = x^2 + x^2 \), which simplifies to \( 256 = 2x^2 \).
4. Solve for \( x^2 \) by dividing both sides of the equation by 2: \( x^2 = \frac{256}{2} \).
5. Find \( x \) by taking the square root of both sides: \( x = \sqrt{128} \). Simplify the square root to express the leg length in simplest radical form.