Okay, so I need to find the area of an isosceles right triangle where the altitude to the hypotenuse is 4√2. Hmm, let's start by recalling some properties of isosceles right triangles. In such a triangle, the legs are equal in length, and the angles are 45°, 45°, and 90°. The hypotenuse can be found using the Pythagorean theorem since the legs are equal. Let me denote the legs as 'a' each, so the hypotenuse would be a√2.

Now, the problem mentions an altitude to the hypotenuse. In a right triangle, the altitude to the hypotenuse has a special relationship with the sides. I remember that in any right triangle, the length of the altitude (h) to the hypotenuse can be calculated using the formula h = (product of the legs) / hypotenuse. Since this is an isosceles right triangle, both legs are equal, so h = (a * a) / (a√2) = a² / (a√2) = a / √2. 

But wait, the problem states that this altitude is 4√2. So, setting up the equation: a / √2 = 4√2. Let me solve for 'a'. Multiply both sides by √2: a = 4√2 * √2. Simplify √2 * √2 = 2, so a = 4 * 2 = 8. Therefore, each leg is 8 units long.

Now, the area of the triangle is (base * height) / 2. Since it's an isosceles right triangle, the base and height are both 'a', so area = (a * a) / 2 = (8 * 8) / 2 = 64 / 2 = 32. 

Wait, but let me double-check this because sometimes when dealing with altitudes in right triangles, there's another way to compute the area. The area can also be calculated as (hypotenuse * altitude) / 2. Let's verify that. The hypotenuse is a√2 = 8√2, and the altitude is 4√2. So, area = (8√2 * 4√2) / 2. Multiply the terms: 8*4 = 32, √2*√2 = 2, so 32*2 = 64. Divide by 2: 64 / 2 = 32. Same result. That confirms the area is 32.

Alternatively, maybe there's another approach using trigonometry. Since it's a right isosceles triangle, the angles are 45°, so the legs are equal, and the altitude to the hypotenuse divides the triangle into two smaller similar triangles. Each of these smaller triangles is also isosceles right triangles. Maybe using similarity ratios? Let's see.

The original triangle has legs 'a' and hypotenuse a√2. The altitude to the hypotenuse splits the hypotenuse into two equal segments because it's an isosceles right triangle. Wait, is that true? Actually, in an isosceles right triangle, the altitude to the hypotenuse also acts as a median and an angle bisector, so it does split the hypotenuse into two equal parts. Therefore, each segment of the hypotenuse would be (a√2)/2 = a/√2. 

But since the altitude is given as 4√2, and in the original triangle, the altitude h = a / √2, which we already found. So that gives a = 8, leading to the same area. 

Another way to confirm: if the legs are 8, then the hypotenuse is 8√2. The area is (8 * 8)/2 = 32. The altitude to the hypotenuse can be found by area = (hypotenuse * altitude)/2 => 32 = (8√2 * h)/2 => 32 = 4√2 * h => h = 32 / (4√2) = 8 / √2 = 4√2. Which matches the given altitude. So all checks out.

Therefore, the area of the triangle is 32.
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