1. **Problem 7: Find the diagonal $x$ of a rectangle with length 24 and width 7.** 2. The diagonal of a rectangle can be found using the Pythagorean theorem: $$x = \sqrt{\text{length}^2 + \text{width}^2}$$ 3. Substitute the given values: $$x = \sqrt{24^2 + 7^2}$$ 4. Calculate the squares: $$x = \sqrt{576 + 49}$$ 5. Add inside the square root: $$x = \sqrt{625}$$ 6. Find the square root: $$x = 25$$ 7. **Answer:** The diagonal $x$ of the rectangle is 25. 8. **Problem 9: Find the diagonal $x$ of a square with side length 8.** 9. The diagonal of a square is given by: $$x = \sqrt{2} \times \text{side}$$ 10. Substitute the side length: $$x = \sqrt{2} \times 8$$ 11. Simplify: $$x = 8\sqrt{2}$$ 12. **Answer:** The diagonal $x$ of the square is $8\sqrt{2}$. 13. **Problem 10: Find the height $x$ of an isosceles triangle with two equal sides 5 and base 8.** 14. The height $x$ splits the base into two equal segments of length 4 each. 15. Use the Pythagorean theorem on one right triangle formed: $$x = \sqrt{5^2 - 4^2}$$ 16. Calculate the squares: $$x = \sqrt{25 - 16}$$ 17. Subtract inside the root: $$x = \sqrt{9}$$ 18. Find the square root: $$x = 3$$ 19. **Answer:** The height $x$ of the isosceles triangle is 3.