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### Problem 8:
**Question: What is the area of the rectangle below if the diagonal has a length of 12 ft?**
Given: Diagonal = 12 ft, one side = 11 ft.

Using the Pythagorean theorem:
\[ \text{Diagonal}^2 = \text{Side 1}^2 + \text{Side 2}^2 \]
\[ 12^2 = 11^2 + \text{Side 2}^2 \]
\[ 144 = 121 + \text{Side 2}^2 \]
\[ \text{Side 2}^2 = 23 \]
\[ \text{Side 2} = \sqrt{23} \]

Area of the rectangle:
\[ \text{Area} = \text{Side 1} \times \text{Side 2} \]
\[ \text{Area} = 11 \times \sqrt{23} \]

Correct answer: (a) \( 11 \sqrt{23} \, \text{ft}^2 \)

### Problem 9:
**Question: A 41-foot ladder is leaned against a wall. If the base of the ladder is 9 ft from the wall, how far above the ground does the ladder touch the wall?**
Given: Ladder length = 41 ft, base distance = 9 ft.

Using the Pythagorean theorem:
\[ \text{Ladder}^2 = \text{Base}^2 + \text{Height}^2 \]
\[ 41^2 = 9^2 + \text{Height}^2 \]
\[ 1681 = 81 + \text{Height}^2 \]
\[ \text{Height}^2 = 1600 \]
\[ \text{Height} = 40 \, \text{ft} \]

Correct answer: (d) 40 ft

### Pythagorean Theorem Practice Set 2:
1. **An isosceles right triangle has legs that measure 10 ft. What is the length of the hypotenuse?**
   Given: Legs = 10 ft each.
   Using Pythagorean theorem:
   \[ \text{Hypotenuse} = \sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2} \]

   Correct answer: (b) \( 10\sqrt{2} \, \text{ft} \)

2. **The legs of a right triangle measure 8 m and 15 m. What is the length of the hypotenuse?**
   Using Pythagorean theorem:
   \[ \text{Hypotenuse} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \, \text{m} \]

   Correct answer: (c) 17 m

3. **What is the missing side length in the right triangle below?**
   Given: \( \text{Hypotenuse} = \sqrt{17} \), one side = \( \sqrt{8} \).

   Using Pythagorean theorem:
   \[ (\sqrt{17})^2 = (\sqrt{8})^2 + \text{Side}^2 \]
   \[ 17 = 8 + \text{Side}^2 \]
   \[ \text{Side}^2 = 9 \]
   \[ \text{Side} = 3 \]

   Correct answer: (a) 3

Would you like details on any of these solutions or have any other questions?

Here are 5 questions you might ask next:
1. How do I solve a quadratic equation?
2. Can you explain the concept of derivatives in calculus?
3. How do I find the integral of a function?
4. What is the Pythagorean theorem and how is it used?
5. Can you help me understand matrix multiplication?

**Tip:** When using the Pythagorean theorem, always ensure to correctly identify the sides of the right triangle, especially distinguishing between the hypotenuse and the legs.

Explore how to solve geometry problems using the Pythagorean theorem with detailed examples. This includes calculating areas of rectangles and determining heights using ladder placements. Suitable for students in Grades 7-9.

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