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In the image, the problem involves solving a right triangle \( \triangle HGF \) with the following information given:

- \( \angle F = 22^\circ \) (this is one of the acute angles)
- \( GF = 8 \) (this is the length of one of the legs)
- \( G \) is the right angle, making \( HF \) the hypotenuse.

You are asked to find the following:
- \( GH \) (the other leg of the triangle)
- \( FG \) (this is already known as 8)
- \( m\angle H \) (the other acute angle)

### Step-by-step solution:

1. **Find \( GH \) using the tangent function:**
   - We know that:
     \[
     \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
     \]
     For \( \theta = 22^\circ \), the opposite side is \( GH \) and the adjacent side is \( FG = 8 \). So:
     \[
     \tan(22^\circ) = \frac{GH}{8}
     \]
     Solving for \( GH \):
     \[
     GH = 8 \times \tan(22^\circ)
     \]
     Using a calculator:
     \[
     GH \approx 8 \times 0.4040 = 3.2
     \]
     So, \( GH \approx 3.2 \).

2. **Find \( HF \) (the hypotenuse) using the cosine function:**
   - The cosine of angle \( F = 22^\circ \) is:
     \[
     \cos(22^\circ) = \frac{GF}{HF}
     \]
     So:
     \[
     HF = \frac{GF}{\cos(22^\circ)} = \frac{8}{\cos(22^\circ)}
     \]
     Using a calculator:
     \[
     HF \approx \frac{8}{0.9272} = 8.63
     \]
     So, \( HF \approx 8.6 \).

3. **Find \( m\angle H \):**
   Since the sum of the angles in a triangle is always \( 180^\circ \), and we know one angle is \( 90^\circ \) and \( \angle F = 22^\circ \), we can calculate \( \angle H \):
   \[
   m\angle H = 90^\circ - 22^\circ = 68^\circ
   \]

### Final answers:
- \( GH \approx 3.2 \)
- \( HF \approx 8.6 \)
- \( m\angle H = 68^\circ \)

Would you like further details or have any other questions?

Here are 5 questions you could explore based on this problem:

1. How would the solution change if the angle was \( 30^\circ \) instead of \( 22^\circ \)?
2. Can you derive the same result using the Pythagorean theorem after finding one leg and the hypotenuse?
3. How would you calculate the area of triangle \( \triangle HGF \)?
4. How would you calculate the perimeter of this triangle?
5. What is the sine of \( \angle F \) in this triangle, and how can it be used to verify the hypotenuse?

**Tip:** Always double-check your trigonometric ratios when using a calculator to avoid rounding errors.

Solve the right triangle where angle F = 22°, FG = 8, and angle G is the right angle. Find GH, FG, and m∠H.

Learn how to solve a right triangle with angle F = 22°, side FG = 8, and angle G = 90°. This problem demonstrates the use of trigonometric ratios to find missing sides and angles, suitable for students in Grades 9-11.

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