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Calculate the length of the third side of the right Triangle in the picture and also its perimeter

To find the length of the third side of the right triangle in the image and its perimeter, we can use the following information from the image:

1. The areas of two squares attached to the legs of the right triangle are given:
   - One square has an area of **50 sq. cm**.
   - The other square has an area of **18 sq. cm**.

### Step 1: Find the side lengths of the squares
Since the area of a square is given by \( \text{Area} = \text{side}^2 \), we can find the side lengths of the squares by taking the square roots of their areas.

- For the square with an area of 50 sq. cm:
  \[
  \text{side} = \sqrt{50} = 7.071 \text{ cm}
  \]

- For the square with an area of 18 sq. cm:
  \[
  \text{side} = \sqrt{18} = 4.243 \text{ cm}
  \]

These are the two sides of the right triangle.

### Step 2: Find the third side of the right triangle (hypotenuse)
The third side is the hypotenuse, and by the **Pythagorean Theorem**:
\[
\text{hypotenuse}^2 = \text{side}_1^2 + \text{side}_2^2
\]
\[
\text{hypotenuse}^2 = 7.071^2 + 4.243^2 = 50 + 18 = 68
\]
\[
\text{hypotenuse} = \sqrt{68} = 8.246 \text{ cm}
\]

### Step 3: Find the perimeter of the right triangle
The perimeter of the triangle is the sum of all three sides:
\[
\text{Perimeter} = 7.071 + 4.243 + 8.246 = 19.560 \text{ cm}
\]

### Final Results:
- **Third side (hypotenuse):** 8.246 cm
- **Perimeter of the triangle:** 19.560 cm

Would you like more details or have any questions?

### Related Questions:
1. What is the Pythagorean theorem, and how is it applied in this case?
2. Can you calculate the area of the triangle given its side lengths?
3. How does the relationship between squares and right triangles work in geometry?
4. How can you verify the accuracy of the perimeter calculation?
5. What are other geometric shapes that involve the Pythagorean theorem?

**Tip:** The Pythagorean theorem only works for right triangles, where one angle is exactly 90°.

Calculate the length of the third side of the right Triangle in the picture and also its perimeter.

Learn how to calculate the hypotenuse and perimeter of a right triangle based on the areas of squares attached to its sides. This geometry problem demonstrates the application of the Pythagorean theorem. Suitable for students in Grades 6-8.

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