Side G 761C30

1. **Problem Statement:** We have a triangle with a horizontal base split into two segments: $g$ on the left and $b$ on the right. The left side of the triangle is length $21$ and forms a $30^\circ$ angle with the base. The right side is length $a$ and forms a $60^\circ$ angle with the base. There is a vertical altitude $h$ from the top vertex to the base. We need to find the exact value of the segment $g$. 2. **Relevant formulas and rules:** - The altitude $h$ can be found using the sine of the angle on either side: $$h = 21 \sin 30^\circ = a \sin 60^\circ$$ - The horizontal segments $g$ and $b$ relate to the sides and angles by cosine: $$g = 21 \cos 30^\circ$$ $$b = a \cos 60^\circ$$ - The base is the sum of $g$ and $b$: $$g + b = \text{base}$$ 3. **Calculate $h$ from the left side:** $$h = 21 \sin 30^\circ = 21 \times \frac{1}{2} = 10.5$$ 4. **Calculate $a$ using $h$ and right side angle:** $$h = a \sin 60^\circ \Rightarrow a = \frac{h}{\sin 60^\circ} = \frac{10.5}{\frac{\sqrt{3}}{2}} = 10.5 \times \frac{2}{\sqrt{3}} = \frac{21}{\sqrt{3}}$$ 5. **Calculate $g$ using left side and cosine:** $$g = 21 \cos 30^\circ = 21 \times \frac{\sqrt{3}}{2} = \frac{21 \sqrt{3}}{2}$$ 6. **Final answer:** $$\boxed{g = \frac{21 \sqrt{3}}{2}}$$ This is the exact value of side $g$ in simplest radical form.