Let's solve the problem in Image 1, step by step, using the approach from Image 2.

Problem (Image 1):

You are dealing with a 45°-45°-90° special right triangle, where one leg is given as $6$, and we need to find the other leg $x$.

Key Properties of a 45°-45°-90° Triangle:

In a 45°-45°-90° triangle, the sides are in the ratio: $1 : 1 : \sqrt{2}$ Where the ratio corresponds to:

• Both legs: $x$
• Hypotenuse: $x\sqrt{2}$

Given Information:

• One leg is $6$.
• The other leg is also $x$.
• The hypotenuse is the leg times $\sqrt{2}$.

Solution:

1. Since both legs are equal in a 45°-45°-90° triangle, the unknown leg is also $6$.
2. To check: $\text{Hypotenuse} = 6 \times \sqrt{2}$

Thus, the solution for $x$ is 6.

Steps Compared to Image 2 Method:

• This problem uses the same idea of ratio scaling based on special triangles.
• Instead of a 30°-60°-90° triangle, the 45°-45°-90° ratio simplifies since both legs are equal.

Would you like further details or clarifications?

Follow-Up Questions:

1. What are the ratios for a 30°-60°-90° triangle?
2. How can you calculate the hypotenuse in a 45°-45°-90° triangle?
3. If the hypotenuse is given, how would you find the legs?
4. How are radical expressions simplified in triangle problems?
5. How do special right triangles differ from general triangles in terms of solving for sides?

Tip: In problems involving right triangles, always check if they are special triangles (45°-45°-90° or 30°-60°-90°) to simplify calculations.