Find G Ef6A4C

1. **State the problem:** We have a triangle with a left side length of $21$, a left base angle of $30^\circ$, a right base angle of $60^\circ$, and the base split into two segments $g$ and $b$. We need to find the exact value of $g$. 2. **Identify the triangle type and properties:** The triangle has angles $30^\circ$, $60^\circ$, and $90^\circ$ (since the altitude $h$ is perpendicular to the base). This is a $30^\circ$-$60^\circ$-$90^\circ$ right triangle. 3. **Recall the side ratios in a $30^\circ$-$60^\circ$-$90^\circ$ triangle:** The sides opposite these angles are in the ratio: $$1 : \sqrt{3} : 2$$ where the side opposite $30^\circ$ is the shortest side, opposite $60^\circ$ is $\sqrt{3}$ times the shortest side, and the hypotenuse is twice the shortest side. 4. **Assign sides based on the given triangle:** The left side of length $21$ is opposite the $60^\circ$ angle (since the left base angle is $30^\circ$, the side opposite it is $g$, and the side opposite $60^\circ$ is $21$). 5. **Find the shortest side (opposite $30^\circ$):** Let the shortest side be $x$. Then: $$21 = x \sqrt{3} \implies x = \frac{21}{\sqrt{3}}$$ 6. **Simplify $x$ by rationalizing the denominator:** $$x = \frac{21}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{21 \sqrt{3}}{3} = 7 \sqrt{3}$$ 7. **Find $g$, the side opposite $30^\circ$:** Since $g$ is the side opposite $30^\circ$, it equals $x$: $$g = 7 \sqrt{3}$$ **Final answer:** $$\boxed{g = 7 \sqrt{3}}$$