1. Problem: Find the missing side (square) in a triangle with sides 3 (left), 6 (right), and 12 (bottom). Formula: Use the Pythagorean theorem for right triangles: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse. Step 1: Identify the hypotenuse. Here, 12 is the longest side, so $c=12$. Step 2: Let the missing side be $x$. Then, $$3^2 + 6^2 = 12^2$$ Step 3: Calculate squares: $$9 + 36 = 144$$ Step 4: Sum left side: $$45 = 144$$ which is false, so the triangle is not right angled or the square is not a side. Since the problem is ambiguous, assume the square is the hypotenuse. Step 5: Let $x$ be the hypotenuse, then $$3^2 + 6^2 = x^2$$ Step 6: Calculate: $$9 + 36 = x^2$$ Step 7: $$45 = x^2$$ Step 8: $$x = \sqrt{45} = 3\sqrt{5} \approx 6.708$$ None of the options match exactly, but closest is 6 (option B: 36 is $6^2$), so answer is B. 36. 2. Problem: Triangle with sides 4 (left), square (right), and 27 (bottom). Step 1: Let the square side be $x$. Step 2: Use Pythagorean theorem: $$4^2 + x^2 = 27^2$$ Step 3: Calculate: $$16 + x^2 = 729$$ Step 4: $$x^2 = 729 - 16 = 713$$ Step 5: $$x = \sqrt{713} \approx 26.7$$ No option matches exactly, but closest is 27 (option B: 9 is too small), so answer is B. 9. 3. Problem: Triangle with sides square (left), 7 (right), and 65 (bottom). Step 1: Let square side be $x$. Step 2: Use Pythagorean theorem: $$x^2 + 7^2 = 65^2$$ Step 3: Calculate: $$x^2 + 49 = 4225$$ Step 4: $$x^2 = 4225 - 49 = 4176$$ Step 5: $$x = \sqrt{4176} \approx 64.64$$ Closest option is A. 80, but 64.64 is closer to 68 (option C), so answer is C. 68. 4. Problem: Triangle with sides 7 (left), 11 (right), and 80 (bottom). Step 1: Check if 80 is hypotenuse. Step 2: $$7^2 + 11^2 = 49 + 121 = 170$$ Step 3: $$80^2 = 6400$$ Not equal, so 80 is not hypotenuse. Step 4: Check if 11 is hypotenuse. Step 5: $$7^2 + 80^2 = 49 + 6400 = 6449$$ Step 6: $$11^2 = 121$$ No match. Step 7: Check if 7 is hypotenuse. Step 8: $$11^2 + 80^2 = 121 + 6400 = 6521$$ Step 9: $$7^2 = 49$$ No match. No right triangle, so answer is the square below bottom side, likely the missing side. Options suggest answer is B. 6. 5. Problem: Triangle with sides 1/2 (left), square (right), and 30 (bottom). Step 1: Let square side be $x$. Step 2: Use Pythagorean theorem: $$(\frac{1}{2})^2 + x^2 = 30^2$$ Step 3: Calculate: $$\frac{1}{4} + x^2 = 900$$ Step 4: $$x^2 = 900 - \frac{1}{4} = 899.75$$ Step 5: $$x = \sqrt{899.75} \approx 29.99$$ Closest option is A. 30. 6. Problem: Triangle with sides 1/6 (left), square (right), and 7 (bottom). Step 1: Let square side be $x$. Step 2: Use Pythagorean theorem: $$(\frac{1}{6})^2 + x^2 = 7^2$$ Step 3: Calculate: $$\frac{1}{36} + x^2 = 49$$ Step 4: $$x^2 = 49 - \frac{1}{36} = 48.9722$$ Step 5: $$x = \sqrt{48.9722} \approx 6.999$$ Closest option is D. 6. 7. Problem: Triangle with sides square (left), 3 (right), and 12 (bottom). Step 1: Let square side be $x$. Step 2: Use Pythagorean theorem: $$x^2 + 3^2 = 12^2$$ Step 3: Calculate: $$x^2 + 9 = 144$$ Step 4: $$x^2 = 135$$ Step 5: $$x = \sqrt{135} = 3\sqrt{15} \approx 11.62$$ Closest option is B. 12. 8. Problem: Triangle with sides 15 (left), square (right), and 0 (bottom). Step 1: Let square side be $x$. Step 2: Use Pythagorean theorem: $$15^2 + x^2 = 0^2$$ Step 3: Calculate: $$225 + x^2 = 0$$ Step 4: $$x^2 = -225$$ No real solution, so answer is B. 0. 9. Problem: Triangle with sides 1/2 (left), square (right), and 1/2 (bottom). Step 1: Let square side be $x$. Step 2: Use Pythagorean theorem: $$(\frac{1}{2})^2 + x^2 = (\frac{1}{2})^2$$ Step 3: Calculate: $$\frac{1}{4} + x^2 = \frac{1}{4}$$ Step 4: $$x^2 = 0$$ Step 5: $$x = 0$$ Closest option is E. 1/2. 10. Problem: Triangle with sides 1 (left), square (right), and 2 (bottom). Step 1: Let square side be $x$. Step 2: Use Pythagorean theorem: $$1^2 + x^2 = 2^2$$ Step 3: Calculate: $$1 + x^2 = 4$$ Step 4: $$x^2 = 3$$ Step 5: $$x = \sqrt{3} \approx 1.732$$ Closest option is B. 4. Final answers: 1: B 2: B 3: C 4: B 5: A 6: D 7: B 8: B 9: E 10: B