1. Problem 1: Greg needs to place a 24-foot ladder so that it reaches 13 feet up the chimney. We need to find how far from the base of the house the ladder should be placed. 2. The Pythagorean theorem states that in a right triangle, $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse (the ladder), and $a$ and $b$ are the legs (height and distance from the house). 3. Substitute the known values: $$13^2 + b^2 = 24^2$$ 4. Calculate squares: $$169 + b^2 = 576$$ 5. Solve for $b^2$: $$b^2 = 576 - 169$$ 6. Simplify: $$b^2 = 407$$ 7. Find $b$ by taking the square root: $$b = \sqrt{407} \approx 20.174$$(rounded to 3 decimals) 8. Round to the nearest tenth: $$b \approx 20.2$$ feet. --- 9. Problem 2: A 72-inch TV has a square screen. We want to find the length and width of the screen. 10. Since the screen is square, length = width = $s$. The diagonal is 72 inches. 11. Using the Pythagorean theorem: $$s^2 + s^2 = 72^2$$ 12. Simplify: $$2s^2 = 5184$$ 13. Solve for $s^2$: $$s^2 = \frac{5184}{2} = 2592$$ 14. Find $s$: $$s = \sqrt{2592} \approx 50.911$$ inches 15. Round to the nearest tenth: $$s \approx 50.9$$ inches. Final answers: - Distance from base of house: 20.2 feet - Length and width of TV screen: 50.9 inches each