1. **Simplify each radical expression.** We use the rule \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \) and rationalize denominators if needed. - \( \frac{15}{\sqrt{3}} = 15 \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{15\sqrt{3}}{3} = 5\sqrt{3} \) - \( \frac{\sqrt{21}}{\sqrt{7}} = \sqrt{\frac{21}{7}} = \sqrt{3} \) - \( \frac{\sqrt{5}}{\sqrt{2}} = \sqrt{\frac{5}{2}} \) (already simplest radical form) - \( \frac{3}{\sqrt{2}} = 3 \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \) 2. **Classify each triangle by sides and angles.** - Triangle with sides 7 m, \( \sqrt{15} \) m, 10 m: - Check if right triangle: \(7^2 + (\sqrt{15})^2 = 49 + 15 = 64\), \(10^2 = 100\), not equal, so not right. - Sides all different, so scalene. - Angles: since no right angle, and sides differ, scalene acute or obtuse. - Triangle with sides 10 m, 8 m, 6 m: - Check right triangle: \(6^2 + 8^2 = 36 + 64 = 100\), \(10^2 = 100\), equal, so right triangle. - Sides all different, scalene. - Triangle with sides 2, 3, \( \sqrt{21} \): - Check right triangle: \(2^2 + 3^2 = 4 + 9 = 13\), \( (\sqrt{21})^2 = 21\), not equal, no right angle. - Sides all different, scalene. - Triangle with sides 8, 8, 12: - Two sides equal, isosceles. - Check right triangle: \(8^2 + 8^2 = 64 + 64 = 128\), \(12^2 = 144\), not equal, no right angle. 3. **Find the missing side in each right triangle.** Use Pythagorean theorem: \(a^2 + b^2 = c^2\) where \(c\) is hypotenuse. - Triangle with legs \(\sqrt{17}\) in and 8 in, hypotenuse \(x\): \[ x^2 = (\sqrt{17})^2 + 8^2 = 17 + 64 = 81 \] \[ x = \sqrt{81} = 9 \text{ in} \] - Triangle with leg 6 yd, hypotenuse 15 yd, base \(x\): \[ x^2 + 6^2 = 15^2 \] \[ x^2 + 36 = 225 \] \[ x^2 = 225 - 36 = 189 \] \[ x = \sqrt{189} = \sqrt{9 \times 21} = 3\sqrt{21} \text{ yd} \] - Triangle with segments 44, 16, 22, and unknown \(x\) with altitude drawn: This is a right triangle split by altitude. Use geometric mean theorem: \[ x^2 = 16 \times 44 = 704 \] \[ x = \sqrt{704} = \sqrt{64 \times 11} = 8\sqrt{11} \] **Final answers:** 1. - \( \frac{15}{\sqrt{3}} = 5\sqrt{3} \) - \( \frac{\sqrt{21}}{\sqrt{7}} = \sqrt{3} \) - \( \frac{\sqrt{5}}{\sqrt{2}} = \sqrt{\frac{5}{2}} \) - \( \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \) 2. - Scalene, obtuse (7, \(\sqrt{15}\), 10) - Scalene, right (10, 8, 6) - Scalene, obtuse (2, 3, \(\sqrt{21}\)) - Isosceles, obtuse (8, 8, 12) 3. - \( x = 9 \) in - \( x = 3\sqrt{21} \) yd - \( x = 8\sqrt{11} \)