1. **State the problem:** Determine if the given side lengths form a triangle and classify the triangle by sides and angles using the Pythagorean theorem and triangle inequality. 2. **Recall the Pythagorean theorem:** For a triangle with sides $a$, $b$, and hypotenuse $c$, if $$a^2 + b^2 = c^2,$$ the triangle is right-angled. 3. **Triangle inequality rule:** The sum of any two sides must be greater than the third side for a valid triangle. 4. **Classification by sides:** - Equilateral: all sides equal - Isosceles: two sides equal - Scalene: all sides different 5. **Classification by angles:** - Right: satisfies Pythagorean theorem - Acute: $a^2 + b^2 > c^2$ - Obtuse: $a^2 + b^2 < c^2$ --- ### Problem 1: Sides 5, 5, $5\sqrt{2}$ - Check triangle inequality: $5 + 5 > 5\sqrt{2}$? $10 > 7.07$ true - Check Pythagorean theorem: $5^2 + 5^2 = 25 + 25 = 50$, $\left(5\sqrt{2}\right)^2 = 25 \times 2 = 50$ - Since $a^2 + b^2 = c^2$, triangle is right-angled. - Two sides equal, so isosceles right triangle. ### Problem 2: Sides $\sqrt{51}$, 11, 13 - Check triangle inequality: $\sqrt{51} + 11 > 13$? $7.14 + 11 = 18.14 > 13$ true - Check Pythagorean theorem: $\left(\sqrt{51}\right)^2 + 11^2 = 51 + 121 = 172$, $13^2 = 169$ - Since $172 > 169$, triangle is acute. - All sides different, scalene acute triangle. ### Problem 3: Sides 13, 12, 9 - Triangle inequality: $12 + 9 = 21 > 13$ true - Pythagorean check: $9^2 + 12^2 = 81 + 144 = 225$, $13^2 = 169$ - Since $225 > 169$, triangle is acute. - All sides different, scalene acute triangle. ### Problem 4: Sides 13, 12, 5 - Triangle inequality: $12 + 5 = 17 > 13$ true - Pythagorean check: $5^2 + 12^2 = 25 + 144 = 169$, $13^2 = 169$ - $a^2 + b^2 = c^2$, right triangle. - All sides different, scalene right triangle. ### Problem 5: Sides 7, 15, 9 - Triangle inequality: $7 + 9 = 16 > 15$ true - Pythagorean check: $7^2 + 9^2 = 49 + 81 = 130$, $15^2 = 225$ - Since $130 < 225$, triangle is obtuse. - All sides different, scalene obtuse triangle. ### Problem 6: Sides 13, 12, 2 - Triangle inequality: $12 + 2 = 14 > 13$ true - Pythagorean check: $2^2 + 12^2 = 4 + 144 = 148$, $13^2 = 169$ - Since $148 < 169$, triangle is obtuse. - All sides different, scalene obtuse triangle. --- ### Problem 7: Sides 5, 12, 13 - Check Pythagorean: $5^2 + 12^2 = 25 + 144 = 169$, $13^2 = 169$ - Right triangle, scalene. ### Problem 8: Sides 9, 9, 15 - Triangle inequality: $9 + 9 = 18 > 15$ true - Pythagorean: $9^2 + 9^2 = 81 + 81 = 162$, $15^2 = 225$ - $162 < 225$, obtuse isosceles triangle. ### Problem 9: Sides 2, $\sqrt{7}$, $\sqrt{11}$ - Triangle inequality: $2 + \sqrt{7} > \sqrt{11}$? $2 + 2.65 = 4.65 > 3.32$ true - Pythagorean check: $2^2 + (\sqrt{7})^2 = 4 + 7 = 11$, $(\sqrt{11})^2 = 11$ - Right triangle, scalene. ### Problem 10: Sides 7, 5, $\sqrt{61}$ - Triangle inequality: $7 + 5 = 12 > \sqrt{61} \approx 7.81$ true - Pythagorean check: $5^2 + 7^2 = 25 + 49 = 74$, $(\sqrt{61})^2 = 61$ - $74 > 61$, acute scalene triangle. ### Problem 11: Sides 12, $\sqrt{29}$, 15 - Triangle inequality: $12 + \sqrt{29} > 15$? $12 + 5.39 = 17.39 > 15$ true - Pythagorean check: $(\sqrt{29})^2 + 12^2 = 29 + 144 = 173$, $15^2 = 225$ - $173 < 225$, obtuse scalene triangle. ### Problem 12: Sides $\frac{1}{2}$, $\frac{7}{2}$, 3 - Triangle inequality: $\frac{1}{2} + 3 = 3.5 > \frac{7}{2} = 3.5$ equal, so no triangle (sum must be strictly greater). --- ### Problem 13: Find $x$ with sides $x$, $\sqrt{6}$, 9 - Assume 9 is hypotenuse. - Use Pythagorean theorem: $$x^2 + (\sqrt{6})^2 = 9^2$$ - $$x^2 + 6 = 81$$ - $$x^2 = 75$$ - $$x = \sqrt{75} = 5\sqrt{3}$$ ### Problem 14: Sides 10, 15, $x$ - Assume 15 is hypotenuse. - $$10^2 + x^2 = 15^2$$ - $$100 + x^2 = 225$$ - $$x^2 = 125$$ - $$x = 5\sqrt{5}$$ ### Problem 15: Sides 8, 14, $x$ - Assume 14 is hypotenuse. - $$8^2 + x^2 = 14^2$$ - $$64 + x^2 = 196$$ - $$x^2 = 132$$ - $$x = 2\sqrt{33}$$ ### Problem 16: Sides $x$, $\sqrt{5}$, $\sqrt{15}$ - Assume $\sqrt{15}$ is hypotenuse. - $$x^2 + (\sqrt{5})^2 = (\sqrt{15})^2$$ - $$x^2 + 5 = 15$$ - $$x^2 = 10$$ - $$x = \sqrt{10}$$ --- ### Problem 17: Rectangle with sides 12 and 8, find diagonal $d$ - Use Pythagorean theorem: $$d^2 = 12^2 + 8^2 = 144 + 64 = 208$$ - $$d = \sqrt{208} = 4\sqrt{13}$$ ### Problem 18: Ladder 20 ft, base 5 ft from wall, find height $h$ - $$h^2 + 5^2 = 20^2$$ - $$h^2 + 25 = 400$$ - $$h^2 = 375$$ - $$h = 5\sqrt{15}$$