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To solve for \( x \) in this triangle, we can use the trigonometric properties of the given angles and lengths. The triangle \( EFG \) has a right angle at \( F \), and we know the lengths and angles involved.

Given:
- \( \angle EFG = 35^\circ \)
- \( \angle DGF = 45^\circ \)
- \( EG = 99 \) cm

We need to find the length of \( x \), which is \( DG \).

Here's a step-by-step solution:

1. **Use the sine rule in triangle \( EFG \):**

   Since \( \triangle EFG \) is a right triangle at \( F \):

   \[
   \sin(35^\circ) = \frac{FG}{EG}
   \]

   Solving for \( FG \):

   \[
   FG = EG \cdot \sin(35^\circ)
   \]

   \[
   FG = 99 \cdot \sin(35^\circ)
   \]

   \[
   FG \approx 99 \cdot 0.5736
   \]

   \[
   FG \approx 56.79 \text{ cm}
   \]

2. **Use the tangent rule in triangle \( DFG \):**

   Since \( \triangle DFG \) is a right triangle at \( F \):

   \[
   \tan(45^\circ) = \frac{DF}{FG}
   \]

   Solving for \( DF \):

   \[
   DF = FG \cdot \tan(45^\circ)
   \]

   \[
   DF = 56.79 \cdot 1
   \]

   \[
   DF \approx 56.79 \text{ cm}
   \]

3. **Calculate the hypotenuse \( x \) in triangle \( DGF \):**

   Using the Pythagorean theorem in \( \triangle DGF \):

   \[
   x^2 = DF^2 + FG^2
   \]

   \[
   x^2 = (56.79)^2 + (56.79)^2
   \]

   \[
   x^2 = 2 \cdot (56.79)^2
   \]

   \[
   x^2 = 2 \cdot 3223.6641
   \]

   \[
   x^2 = 6447.3282
   \]

   \[
   x \approx \sqrt{6447.3282}
   \]

   \[
   x \approx 80.29 \text{ cm}
   \]

Thus, the value of \( x \) to the nearest centimeter is \( 80 \) cm.

Would you like any further details or have any questions?

Here are some questions you might consider next:
1. How to find the length of the hypotenuse in a right triangle?
2. What is the sine rule and how is it applied in solving triangle problems?
3. How to use the Pythagorean theorem to solve for an unknown side in a right triangle?
4. How can we apply trigonometric ratios to find unknown lengths in right triangles?
5. How to solve problems involving special angles like 45° and 30° in trigonometry?

**Tip:** Always check the angles and side lengths you have and see if any trigonometric rules or properties can be applied to simplify your calculations.

Learn how to calculate the length of DG in triangle EFG using trigonometric properties. Detailed step-by-step solution involving the sine rule, tangent rule, and the Pythagorean theorem. Suitable for high school students in Grades 10-12.

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