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Correction exercice math en francais merci 

To solve this exercise, let's calculate the lengths \( HG \) and \( EF \) in the given right triangle using the Pythagorean theorem and properties of similar triangles.

### Observations from the Figure
1. We have a right triangle \( \triangle EFG \) with a right angle at \( H \).
2. \( EH = 2 \) (horizontal segment).
3. \( HF = 8 \) (vertical segment).
4. \( FG = 10 \) (hypotenuse of \( \triangle HFG \)).

#### Step 1: Calculating \( HG \)
Since \( \triangle HFG \) is a right triangle with \( HF = 8 \) and \( HG = x \) as the unknown, and \( FG = 10 \) as the hypotenuse, we can apply the Pythagorean theorem:

\[
HG^2 + HF^2 = FG^2
\]

Substitute the known values:

\[
HG^2 + 8^2 = 10^2
\]

\[
HG^2 + 64 = 100
\]

\[
HG^2 = 100 - 64
\]

\[
HG^2 = 36
\]

\[
HG = \sqrt{36} = 6
\]

So, \( HG = 6 \).

#### Step 2: Calculating \( EF \)
The segment \( EF \) can be found using the similarity of triangles \( \triangle EFG \) and \( \triangle HFG \). Since \( \triangle HFG \) is a smaller, similar triangle with \( EF \) as the corresponding side of \( FG \), we can set up a proportion.

Since \( EH = 2 \), \( HG = 6 \), and \( FG = 10 \), the triangle proportions give:

\[
\frac{EF}{FG} = \frac{EH}{HG}
\]

Substitute the known values:

\[
\frac{EF}{10} = \frac{2}{6}
\]

Solve for \( EF \):

\[
EF = 10 \times \frac{2}{6}
\]

\[
EF = 10 \times \frac{1}{3} = \frac{10}{3} \approx 3.33
\]

So, \( EF \approx 3.33 \) (or exactly \( \frac{10}{3} \)).

### Final Answer
- \( HG = 6 \)
- \( EF = \frac{10}{3} \) (approximately \( 3.33 \))

Would you like further details on any part of the solution?

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Here are 5 related questions for further practice:
1. How would you find the length of \( EF \) if only \( FG \) and the angle \( \angle EFG \) were given?
2. Can you find the area of \( \triangle EFG \)?
3. What is the perimeter of \( \triangle EFG \)?
4. How would the calculations change if \( \triangle EFG \) were not a right triangle?
5. Can you verify the similarity between \( \triangle EFG \) and \( \triangle HFG \) using angle-angle similarity?

**Tip:** In right triangles, the Pythagorean theorem is a powerful tool to find missing side lengths when you know two sides.

Observer la figure puis calculer les longueurs HG et EF

Learn how to calculate the lengths of segments HG and EF in a right triangle using the Pythagorean theorem and the properties of similar triangles. This exercise involves geometry concepts such as right triangles, similarity, and proportions, suitable for students in Grades 8-10.

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