https://file.aiquickdraw.com/mathbot/file/1726070984383lkou82pp.jpg

The triangle in the image is a 45°-45°-90° right triangle, which has specific properties that can help us solve for the unknown side \( f \).

### Key Properties of a 45°-45°-90° Triangle:
- In a 45°-45°-90° triangle, the two legs are congruent, and the hypotenuse is \( \sqrt{2} \) times the length of a leg.
- The formula for the sides of a 45°-45°-90° triangle is given by:
  \[
  \text{Hypotenuse} = \text{Leg} \times \sqrt{2}
  \]
  
In this case, the hypotenuse is given as \( \sqrt{2} \) mm, and both legs (denoted by \( f \)) will have the same length.

### Step-by-Step Solution:
1. Let the length of the leg be \( f \).
2. According to the property, the hypotenuse is \( f \times \sqrt{2} \), but here the hypotenuse is given as \( \sqrt{2} \) mm.
   
3. Setting up the equation for the hypotenuse:
   \[
   f \times \sqrt{2} = \sqrt{2}
   \]
   
4. Solving for \( f \):
   \[
   f = \frac{\sqrt{2}}{\sqrt{2}} = 1 \, \text{mm}
   \]

Thus, the length of \( f \) is \( 1 \) mm.

Would you like further details or have any additional questions?

Here are some related questions to consider:
1. What is the general formula for any right triangle's side lengths?
2. How does the 45°-45°-90° triangle differ from a 30°-60°-90° triangle?
3. Can this problem be solved using trigonometric functions?
4. How would the problem change if the hypotenuse were a different value?
5. What are some real-life applications of the 45°-45°-90° triangle?

**Tip:** Memorizing the properties of special triangles like the 45°-45°-90° and 30°-60°-90° triangles can make solving geometry problems much faster!

Find the value of f in a 45°-45°-90° triangle where the hypotenuse is √2 mm.

Learn how to calculate the leg of a 45°-45°-90° triangle when the hypotenuse is √2 mm. This geometry problem explores the properties of special right triangles and uses the formula Hypotenuse = Leg * √2. Suitable for students in Grades 7-10.

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