1. **Problem 1:** Find the length of side $x$ in a right triangle with angles $39^\circ$ and $60^\circ$, side adjacent to $39^\circ$ is 38, side opposite $60^\circ$ is 5, hypotenuse labeled $x$. 2. **Formula:** Use the Pythagorean theorem for right triangles: $$x^2 = a^2 + b^2$$ where $a$ and $b$ are legs, $x$ is hypotenuse. 3. **Calculate:** Given legs 38 and 5, $$x = \sqrt{38^2 + 5^2} = \sqrt{1444 + 25} = \sqrt{1469}$$ 4. **Intermediate value:** $\sqrt{1469} \approx 38.3$ (rounded to nearest tenth). 5. **Answer:** $x \approx 38.3$. 1. **Problem 2:** Right triangle with angles $35^\circ$ and $61^\circ$, side adjacent to $35^\circ$ is 9, hypotenuse labeled $x$. 2. **Formula:** Use cosine definition: $$\cos(35^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{9}{x}$$ 3. **Calculate:** $$x = \frac{9}{\cos(35^\circ)}$$ 4. **Intermediate value:** $\cos(35^\circ) \approx 0.8192$, so $$x = \frac{9}{0.8192} \approx 11.0$$ 5. **Answer:** $x \approx 11.0$. 1. **Problem 3:** Triangle with sides 23 and unknown side $x$, angles $101^\circ$ and $50^\circ$. 2. **Formula:** Use Law of Sines: $$\frac{x}{\sin(101^\circ)} = \frac{23}{\sin(50^\circ)}$$ 3. **Calculate:** $$x = \frac{23 \times \sin(101^\circ)}{\sin(50^\circ)}$$ 4. **Intermediate values:** $\sin(101^\circ) \approx 0.9816$, $\sin(50^\circ) \approx 0.7660$ $$x = \frac{23 \times 0.9816}{0.7660} = \frac{22.577}{0.7660} \approx 29.5$$ 5. **Answer:** $x \approx 29.5$. 1. **Problem 4:** Triangle with sides 12 and unknown side $x$, angles $71^\circ$ and $44^\circ$. 2. **Formula:** Law of Sines: $$\frac{x}{\sin(44^\circ)} = \frac{12}{\sin(71^\circ)}$$ 3. **Calculate:** $$x = \frac{12 \times \sin(44^\circ)}{\sin(71^\circ)}$$ 4. **Intermediate values:** $\sin(44^\circ) \approx 0.6947$, $\sin(71^\circ) \approx 0.9455$ $$x = \frac{12 \times 0.6947}{0.9455} = \frac{8.336}{0.9455} \approx 8.8$$ 5. **Answer:** $x \approx 8.8$. 1. **Problem 5:** Find area of right triangle with angles $70^\circ$ and $44^\circ$, side adjacent to $70^\circ$ is 46. 2. **Formula:** Area = $\frac{1}{2}ab$ where $a$ and $b$ are legs. 3. **Calculate other leg:** $$b = a \times \tan(70^\circ) = 46 \times \tan(70^\circ)$$ 4. **Intermediate value:** $\tan(70^\circ) \approx 2.7475$ $$b = 46 \times 2.7475 = 126.4$$ 5. **Area:** $$\frac{1}{2} \times 46 \times 126.4 = 23 \times 126.4 = 2907.2$$ 6. **Check given area 528.6:** The given area does not match calculation; possibly a different side or method was intended. 1. **Problem 6:** Find area of right triangle with angles $50^\circ$ and $65^\circ$, side adjacent to $50^\circ$ is 41. 2. **Calculate other leg:** $$b = 41 \times \tan(50^\circ)$$ 3. **Intermediate value:** $\tan(50^\circ) \approx 1.1918$ $$b = 41 \times 1.1918 = 48.9$$ 4. **Area:** $$\frac{1}{2} \times 41 \times 48.9 = 20.5 \times 48.9 = 1001.0$$ 5. **Check given area 1974.6:** The given area does not match calculation; possibly a different side or method was intended. **Summary of answers:** 1) $x \approx 38.3$ 2) $x \approx 11.0$ 3) $x \approx 29.5$ 4) $x \approx 8.8$ 5) Area $\approx 2907.2$ (discrepancy with given 528.6) 6) Area $\approx 1001.0$ (discrepancy with given 1974.6)