## You're welcome! It's great to hear that you found the answer to your geometry problem using the Pythagorean theorem and ratio rules for triangles.

For the benefit of other learners, let me explain how you can apply these concepts to solve similar problems:

1. The Pythagorean theorem: This theorem is particularly useful for right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. So, if you know the lengths of two sides of a right triangle, you can use the Pythagorean theorem to find the length of the third side.

2. Ratio rules for triangles: Triangles have several important ratios that can be used to find missing side lengths or angles. Here are a few key ones:

- The sine ratio: In a right triangle, the ratio of the length of the side opposite an angle to the length of the hypotenuse is equal to the sine of that angle.

- The cosine ratio: In a right triangle, the ratio of the length of the side adjacent to an angle to the length of the hypotenuse is equal to the cosine of that angle.

- The tangent ratio: In a right triangle, the ratio of the length of the side opposite an angle to the length of the side adjacent to that angle is equal to the tangent of that angle.

By using these ratio rules along with the known values of angles or side lengths, you can set up and solve equations to find missing information about the triangle.

Remember, practice is key when it comes to mastering geometry. Keep exploring various problem-solving techniques and theorems, and you'll continue to improve your skills.