calculate side length of isosceles triangle using law of sines

1. Identify the triangle properties: We are given an isosceles triangle with angles $56^{\circ}$, $56^{\circ}$, and $68^{\circ}$. The side opposite the $68^{\circ}$ angle is $24$ cm. Let the sides opposite the $56^{\circ}$ angles be $x$. Since the triangle is isosceles, these two sides are equal.

1. Apply the Law of Sines: The Law of Sines states that $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$. Here, we can set up the ratio: $\frac{x}{\sin(56^{\circ})} = \frac{24}{\sin(68^{\circ})}$.

3. Solve for x: To find $x$, multiply both sides by $\sin(56^{\circ})$:
$$x = \frac{24 \cdot \sin(56^{\circ})}{\sin(68^{\circ})}$$

4. Calculate the numerical value: Using a calculator:
$\sin(56^{\circ}) \approx 0.8290$
$\sin(68^{\circ}) \approx 0.9272$
$$x \approx \frac{24 \cdot 0.8290}{0.9272} \approx \frac{19.896}{0.9272} \approx 21.458$$

1. Conclusion: Rounding to one decimal place, we get $21.5$ cm, which matches option D.