1. **Problem statement:** Given squares on the sides of triangles with some areas known and some unknown, determine if the triangles are right triangles and find missing areas or side lengths using the Pythagorean theorem. 2. **Formula:** For a right triangle with legs $a$, $b$ and hypotenuse $c$, the Pythagorean theorem states: $$a^2 + b^2 = c^2$$ This means the sum of the areas of the squares on the legs equals the area of the square on the hypotenuse. 3. **Check if triangle with squares 169 cm², 144 cm², and 500 cm² is right angled:** - Sum of smaller squares: $169 + 144 = 313$ - Compare with largest square: $500$ - Since $313 \neq 500$, this is **not** a right triangle. 4. **Find missing side lengths from given areas:** - Side length is the square root of the area. - For 169 cm²: $\sqrt{169} = 13$ cm - For 144 cm²: $\sqrt{144} = 12$ cm - For 500 cm²: $\sqrt{500} = 10\sqrt{5} \approx 22.36$ cm 5. **Calculate missing area for triangle with squares 400 cm², 500 cm², and unknown:** - Let unknown area be $x$ - Assume 400 cm² and $x$ are legs, 500 cm² is hypotenuse - Using Pythagorean theorem: $$400 + x = 500$$ $$x = 500 - 400 = 100$$ - So, missing area is $100$ cm² 6. **Check if triangle with squares 25 cm², 144 cm², and 169 cm² is right angled:** - Sum of smaller squares: $25 + 144 = 169$ - Largest square: $169$ - Since $25 + 144 = 169$, this triangle **is** right angled. 7. **Find side lengths for this triangle:** - $\sqrt{25} = 5$ - $\sqrt{144} = 12$ - $\sqrt{169} = 13$ **Final answers:** - Triangle with squares 169, 144, 500 cm² is not right angled. - Missing area next to 400 cm² and 500 cm² squares is $100$ cm². - Triangle with squares 25, 144, 169 cm² is right angled. - Side lengths for right triangle are 5 cm, 12 cm, and 13 cm.