1. **Problem:** Determine the type of triangle based on the lengths of the sides. 2. **Formula and rules:** - Use the triangle inequality: sum of any two sides must be greater than the third. - To classify: - Equilateral: all sides equal. - Isosceles: two sides equal. - Scalene: all sides different. - Right triangle: satisfies Pythagoras theorem $a^2 + b^2 = c^2$ where $c$ is the longest side. 3. **Check each set:** **Set 1: 5, 3, 6** - Check triangle inequality: $5 + 3 = 8 > 6$, $5 + 6 = 11 > 3$, $3 + 6 = 9 > 5$ (valid triangle) - Check right triangle: Longest side $6$, check $5^2 + 3^2 = 25 + 9 = 34$ vs $6^2 = 36$ (not equal) - All sides different, so scalene. **Set 2: 4, 7, 8** - Triangle inequality: $4 + 7 = 11 > 8$, $4 + 8 = 12 > 7$, $7 + 8 = 15 > 4$ (valid) - Right triangle check: Longest side $8$, $4^2 + 7^2 = 16 + 49 = 65$ vs $8^2 = 64$ (not equal) - All sides different, scalene. **Set 3: 17, 8, 18** - Triangle inequality: $17 + 8 = 25 > 18$, $17 + 18 = 35 > 8$, $8 + 18 = 26 > 17$ (valid) - Right triangle check: Longest side $18$, $17^2 + 8^2 = 289 + 64 = 353$ vs $18^2 = 324$ (not equal) - All sides different, scalene. **Set 4: 13, 8, 6** - Triangle inequality: $13 + 8 = 21 > 6$, $13 + 6 = 19 > 8$, $8 + 6 = 14 > 13$ (valid) - Right triangle check: Longest side $13$, $8^2 + 6^2 = 64 + 36 = 100$ vs $13^2 = 169$ (not equal) - All sides different, scalene. **Set 5: $\sqrt{15}$, $\sqrt{8}$, $\sqrt{7}$** - Triangle inequality: $\sqrt{15} + \sqrt{8} > \sqrt{7}$ (true since $3.87 + 2.83 > 2.65$) $\sqrt{15} + \sqrt{7} > \sqrt{8}$ (true) $\sqrt{8} + \sqrt{7} > \sqrt{15}$ (true) - Right triangle check: Longest side $\sqrt{15}$, check if $\left(\sqrt{8}\right)^2 + \left(\sqrt{7}\right)^2 = \left(\sqrt{15}\right)^2$ $8 + 7 = 15$ equals $15$ (true) - So this is a right triangle. **Final answers:** 1. Scalene 2. Scalene 3. Scalene 4. Scalene 5. Right triangle