1. Problem 5: Given the equation $a^2 + b^2 = c^2$ with $16^2 + x^2 = 25^2$, find $x$. 2. Use the Pythagorean theorem formula: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse (longest side), and $a$, $b$ are the legs. 3. Substitute the known values: $$16^2 + x^2 = 25^2$$ $$256 + x^2 = 625$$ 4. Isolate $x^2$: $$x^2 = 625 - 256$$ $$x^2 = 369$$ 5. Take the square root of both sides: $$x = \sqrt{369}$$ 6. Simplify the square root if possible: $$369 = 3 \times 123 = 3 \times 3 \times 41 = 9 \times 41$$ $$x = \sqrt{9 \times 41} = \sqrt{9} \times \sqrt{41} = 3\sqrt{41}$$ 7. Approximate the decimal value: $$x \approx 3 \times 6.403 = 19.21$$ --- 8. Problem 7: Use a Pythagorean triple to find $x$ in the triangle with sides 8, 17, and $x$. 9. The triple is $(8, 15, 17)$, so $x = 15$. --- 10. Problem 8: Triangle with sides 45, 24, and $x$. 11. Use Pythagorean theorem: $$24^2 + x^2 = 45^2$$ $$576 + x^2 = 2025$$ $$x^2 = 2025 - 576 = 1449$$ $$x = \sqrt{1449}$$ 12. Simplify: $$1449 = 9 \times 161$$ $$x = 3\sqrt{161} \approx 3 \times 12.688 = 38.06$$ --- 13. Problem 9: Triangle with sides 28, 96, and $x$. 14. Use Pythagorean theorem: $$28^2 + 96^2 = x^2$$ $$784 + 9216 = x^2$$ $$10000 = x^2$$ $$x = \sqrt{10000} = 100$$ --- 15. Problem 10: Triangle with sides 5, 12, and $x$. 16. Use Pythagorean theorem: $$5^2 + 12^2 = x^2$$ $$25 + 144 = x^2$$ $$169 = x^2$$ $$x = 13$$ --- 17. Problem 11: Triangle with sides 8, 10, and $x$. 18. Use Pythagorean theorem: $$8^2 + 10^2 = x^2$$ $$64 + 100 = x^2$$ $$164 = x^2$$ $$x = \sqrt{164} = 2\sqrt{41} \approx 12.81$$ --- 19. Problem 12: Triangle with sides 12, 20, and $x$. 20. Use Pythagorean theorem: $$12^2 + 20^2 = x^2$$ $$144 + 400 = x^2$$ $$544 = x^2$$ $$x = \sqrt{544} = 4\sqrt{34} \approx 23.32$$ --- 21. Problem 13: Ramp problem. Base = 10 ft, ramp length (hypotenuse) = 11 ft, find height $h$. 22. Use Pythagorean theorem: $$10^2 + h^2 = 11^2$$ $$100 + h^2 = 121$$ $$h^2 = 121 - 100 = 21$$ $$h = \sqrt{21} \approx 4.58$$ --- 23. Problem 14: Flight problem. East = 60 miles, North = 25 miles, find distance $d$. 24. Use Pythagorean theorem: $$60^2 + 25^2 = d^2$$ $$3600 + 625 = d^2$$ $$4225 = d^2$$ $$d = \sqrt{4225} = 65$$ Final answers: - Problem 5: $x = 3\sqrt{41} \approx 19.21$ - Problem 7: $x = 15$ - Problem 8: $x = 3\sqrt{161} \approx 38.06$ - Problem 9: $x = 100$ - Problem 10: $x = 13$ - Problem 11: $x \approx 12.81$ - Problem 12: $x \approx 23.32$ - Problem 13: $h \approx 4.58$ - Problem 14: $d = 65$