1. **State the problem:** We have a right triangle with hypotenuse $19\sqrt{2}$, one acute angle $45^\circ$, and the other acute angle $30^\circ$. We need to find the lengths of sides $a$, $b$, $c$, and $d$. 2. **Analyze the triangle:** Since the triangle has angles $45^\circ$, $30^\circ$, and $90^\circ$, this is not a standard right triangle because the angles do not sum to $180^\circ$ (they sum to $165^\circ$). However, the problem likely means the triangle is composed of two right triangles or the labels correspond to two different triangles. Given the hypotenuse and angles, we can find sides opposite and adjacent to each angle using trigonometric ratios. 3. **Use trigonometric definitions:** - For angle $45^\circ$: - Opposite side: $c$ - Adjacent side: $a$ - For angle $30^\circ$: - Opposite side: $b$ - Adjacent side: $d$ 4. **Use the hypotenuse to find sides:** Since the hypotenuse is $19\sqrt{2}$, for the $45^\circ$ angle: $$ c = (19\sqrt{2}) \times \sin 45^\circ = 19\sqrt{2} \times \frac{\sqrt{2}}{2} = 19 \times \cancel{\sqrt{2}} \times \frac{\cancel{\sqrt{2}}}{2} = 19 $$ Similarly, for side $a$ (adjacent to $45^\circ$): $$ a = (19\sqrt{2}) \times \cos 45^\circ = 19\sqrt{2} \times \frac{\sqrt{2}}{2} = 19 $$ 5. **For the $30^\circ$ angle:** - Opposite side $b$: $$ b = (19\sqrt{2}) \times \sin 30^\circ = 19\sqrt{2} \times \frac{1}{2} = \frac{19\sqrt{2}}{2} $$ - Adjacent side $d$: $$ d = (19\sqrt{2}) \times \cos 30^\circ = 19\sqrt{2} \times \frac{\sqrt{3}}{2} = \frac{19\sqrt{6}}{2} $$ 6. **Summary of answers:** $$ a = 19 $$ $$ b = \frac{19\sqrt{2}}{2} $$ $$ c = 19 $$ $$ d = \frac{19\sqrt{6}}{2} $$ These are the exact simplified lengths of the sides.