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### Question 1: Find the value of \( x \).

#### Part (a):
- We have a right triangle with sides 3 cm, 7 cm, and \( x \) cm (hypotenuse).
- By the Pythagorean Theorem:
  \[
  x^2 = 3^2 + 7^2
  \]
  \[
  x^2 = 9 + 49
  \]
  \[
  x^2 = 58
  \]
  \[
  x = \sqrt{58} \approx 7.62 \, \text{cm}
  \]

#### Part (b):
- Another right triangle with sides 11 cm, 13 cm, and \( x \) cm (one of the legs this time).
- Again, using the Pythagorean Theorem:
  \[
  13^2 = 11^2 + x^2
  \]
  \[
  169 = 121 + x^2
  \]
  \[
  x^2 = 169 - 121
  \]
  \[
  x^2 = 48
  \]
  \[
  x = \sqrt{48} = 4\sqrt{3} \approx 6.93 \, \text{cm}
  \]

### Question 2: Find the area of the innermost square.
- ABCD is a square with side length 12 cm.
- The squares are inscribed recursively within each other, forming a spiral pattern. This is likely a geometric progression where each square's side length reduces by a constant factor.
  
Let’s assume the reduction factor is \( \frac{\sqrt{2}}{2} \) based on the way squares are inscribed in each other at 45-degree angles (this is a common geometric relationship for inscribed squares).

- The first square has side length 12 cm.
- The second square's side is \( 12 \times \frac{\sqrt{2}}{2} = 12 \times 0.7071 \approx 8.49 \, \text{cm} \).
- Continuing with the same ratio:
  - Third square: \( 8.49 \times 0.7071 \approx 6 \, \text{cm} \),
  - Fourth square: \( 6 \times 0.7071 \approx 4.24 \, \text{cm} \),
  - Fifth square (innermost): \( 4.24 \times 0.7071 \approx 3 \, \text{cm} \).

The area of the innermost square is:
\[
A = 3^2 = 9 \, \text{cm}^2
\]

Would you like further clarification or more details on any part? Here are some related questions for additional practice:

1. What would be the length of the hypotenuse if one leg of a triangle is 6 cm and the other leg is 8 cm?
2. How do you find the area of a square inscribed inside a circle?
3. How can the Pythagorean theorem be applied to find the length of a diagonal in a rectangle?
4. If the reduction factor for the squares was different, how would that affect the area of the innermost square?
5. What is the geometric sequence of side lengths in an inscribed square problem like this one?

**Tip**: Always double-check your triangle sides using the Pythagorean theorem, as it's a quick way to ensure correctness when dealing with right triangles.

1. Find the value of x. 

 a. Given a right triangle with sides 3 cm and 7 cm. 

 b. Given a right triangle with sides 11 cm and 13 cm. 

 2. ABCD is a square with side 12 cm. Find the area of the innermost square.

Learn how to solve for x in right triangles using the Pythagorean theorem and find the area of the innermost square in a recursive geometric pattern. Suitable for students in Grades 8-10, this problem covers key geometric concepts and theorems.

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