1. **State the problem:** We have an isosceles triangle with two equal sides of length $n$ and the angle between these sides is $52^\circ$. The perpendicular height from the vertex opposite the base to the base is $4.2$ mm. We need to find the length $n$ to 1 decimal place. 2. **Understand the triangle:** The perpendicular height splits the isosceles triangle into two right triangles. Each right triangle has: - Hypotenuse = $n$ (one of the equal sides) - One angle = $\frac{52^\circ}{2} = 26^\circ$ (since the height bisects the vertex angle) - Opposite side to $26^\circ$ = height = $4.2$ mm 3. **Use trigonometry:** In the right triangle, the side opposite the angle $26^\circ$ is the height, and the hypotenuse is $n$. Using the sine function: $$\sin(26^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4.2}{n}$$ 4. **Solve for $n$:** $$n = \frac{4.2}{\sin(26^\circ)}$$ 5. **Calculate $\sin(26^\circ)$:** $$\sin(26^\circ) \approx 0.4384$$ 6. **Calculate $n$:** $$n = \frac{4.2}{0.4384} \approx 9.58$$ 7. **Round to 1 decimal place:** $$n \approx 9.6 \text{ mm}$$ **Final answer:** The length $n$ is approximately **9.6 mm**.