1. **Problem Statement:** Find the length labeled $j$ in the given geometric diagram involving circles, arcs, and triangles with angles 26°, 64°, 32°, 45°, and 77°. 2. **Identify the triangle of interest:** The yellow highlighted right triangle near the top-left has angles 26° and 64°, and one side length is given as 11. Since the sum of angles in any triangle is 180°, the third angle is 90°, confirming it is a right triangle. 3. **Label sides:** Let the side opposite 26° be $j$, the side adjacent to 26° be 11, and the hypotenuse be $h$. Since $j$ is opposite the 26° angle, use sine: $$j = h \sin(26^\circ)$$ And since 11 is adjacent to 26°, $$11 = h \cos(26^\circ)$$ 4. **Find the hypotenuse $h$:** Solve $$h = \frac{11}{\cos(26^\circ)}$$ Calculate: $$\cos(26^\circ) \approx 0.8988$$ $$h \approx \frac{11}{0.8988} \approx 12.24$$ 5. **Calculate $j$:** $$j = 12.24 \times \sin(26^\circ)$$ Calculate: $$\sin(26^\circ) \approx 0.4384$$ $$j \approx 12.24 \times 0.4384 \approx 5.37$$ 6. **Interpretation:** The length $j\approx 5.37$ units. **Final answer:** $j \approx 5.37$