Determine if the following lengths make an acute, right or obtuse triangle. 5. 10, 15, 18. 6. 4, 20, 21. 7. 15, 16, 17. 8. $$ 15,15,15 \sqrt{2} $$. 9. 12, 17, 19. 10. 3, 4, 5 .

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To determine whether each set of side lengths forms an **acute**, **right**, or **obtuse** triangle, we'll use the **converse of the Pythagorean theorem**. Here's the step-by-step process: 1. **Arrange the sides in ascending order**: Let $$ a \leq b \leq c $$. 2. **Apply the triangle inequality** to ensure a valid triangle: - $$ a + b > c $$ 3. **Compare $$ a^2 + b^2 $$ with $$ c^2 $$**: - If $$ a^2 + b^2 > c^2 $$, the triangle is **acute**. - If $$ a^2 + b^2 = c^2 $$, the triangle is **right**. - If $$ a^2 + b^2 < c^2 $$, the triangle is **obtuse**. Let's apply this to each set of lengths: --- ### 5. **Sides: 10, 15, 18** - **Ordered**: $$ a = 10 $$, $$ b = 15 $$, $$ c = 18 $$ - **Triangle Inequality**: - $$ 10 + 15 = 25 > 18 $$ ✓ - $$ 10 + 18 = 28 > 15 $$ ✓ - $$ 15 + 18 = 33 > 10 $$ ✓ - **Calculation**: - $$ 10^2 + 15^2 = 100 + 225 = 325 $$ - $$ 18^2 = 324 $$ - $$ 325 > 324 $$ - **Result**: **Acute Triangle** --- ### 6. **Sides: 4, 20, 21** - **Ordered**: $$ a = 4 $$, $$ b = 20 $$, $$ c = 21 $$ - **Triangle Inequality**: - $$ 4 + 20 = 24 > 21 $$ ✓ - $$ 4 + 21 = 25 > 20 $$ ✓ - $$ 20 + 21 = 41 > 4 $$ ✓ - **Calculation**: - $$ 4^2 + 20^2 = 16 + 400 = 416 $$ - $$ 21^2 = 441 $$ - $$ 416 < 441 $$ - **Result**: **Obtuse Triangle** --- ### 7. **Sides: 15, 16, 17** - **Ordered**: $$ a = 15 $$, $$ b = 16 $$, $$ c = 17 $$ - **Triangle Inequality**: - $$ 15 + 16 = 31 > 17 $$ ✓ - $$ 15 + 17 = 32 > 16 $$ ✓ - $$ 16 + 17 = 33 > 15 $$ ✓ - **Calculation**: - $$ 15^2 + 16^2 = 225 + 256 = 481 $$ - $$ 17^2 = 289 $$ - $$ 481 > 289 $$ - **Result**: **Acute Triangle** --- ### 8. **Sides: $$ 15 $$, $$ 15 $$, $$ 15\sqrt{2} $$** - **Ordered**: $$ a = 15 $$, $$ b = 15 $$, $$ c = 15\sqrt{2} $$ (approximately 21.21) - **Triangle Inequality**: - $$ 15 + 15 = 30 > 15\sqrt{2} $$ ✓ - $$ 15 + 15\sqrt{2} \approx 36.21 > 15 $$ ✓ - $$ 15 + 15\sqrt{2} \approx 36.21 > 15 $$ ✓ - **Calculation**: - $$ 15^2 + 15^2 = 225 + 225 = 450 $$ - $$ (15\sqrt{2})^2 = 225 imes 2 = 450 $$ - $$ 450 = 450 $$ - **Result**: **Right Triangle** --- ### 9. **Sides: 12, 17, 19** - **Ordered**: $$ a = 12 $$, $$ b = 17 $$, $$ c = 19 $$ - **Triangle Inequality**: - $$ 12 + 17 = 29 > 19 $$ ✓ - $$ 12 + 19 = 31 > 17 $$ ✓ - $$ 17 + 19 = 36 > 12 $$ ✓ - **Calculation**: - $$ 12^2 + 17^2 = 144 + 289 = 433 $$ - $$ 19^2 = 361 $$ - $$ 433 > 361 $$ - **Result**: **Acute Triangle** --- ### 10. **Sides: 3, 4, 5** - **Ordered**: $$ a = 3 $$, $$ b = 4 $$, $$ c = 5 $$ - **Triangle Inequality**: - $$ 3 + 4 = 7 > 5 $$ ✓ - $$ 3 + 5 = 8 > 4 $$ ✓ - $$ 4 + 5 = 9 > 3 $$ ✓ - **Calculation**: - $$ 3^2 + 4^2 = 9 + 16 = 25 $$ - $$ 5^2 = 25 $$ - $$ 25 = 25 $$ - **Result**: **Right Triangle** --- ### **Summary** 5. **10, 15, 18**: **Acute Triangle** 6. **4, 20, 21**: **Obtuse Triangle** 7. **15, 16, 17**: **Acute Triangle** 8. **15, 15, 15√2**: **Right Triangle** 9. **12, 17, 19**: **Acute Triangle** 10. **3, 4, 5**: **Right Triangle** - **10, 15, 18**: Acute Triangle - **4, 20, 21**: Obtuse Triangle - **15, 16, 17**: Acute Triangle - **15, 15, 15√2**: Right Triangle - **12, 17, 19**: Acute Triangle - **3, 4, 5**: Right Triangle

Determine if the following lengths make an acute, right or obtuse triangle. 5. 10, 15, 18. 6. 4, 20, 21. 7. 15, 16, 17. 8. \( 15,15,15 \sqrt{2} \). 9. 12, 17, 19. 10. 3, 4, 5 .

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