1. **Problem Statement:** We have three polygons: an equilateral triangle, a rhombus with a 60° angle, and a regular hexagon. Each contains mutually tangent congruent disks. We denote by $T$, $R$, and $H$ the ratios of the total area of the disks to the area of the polygon for the triangle, rhombus, and hexagon respectively. We want to determine which relation among $T$, $R$, and $H$ is true from the given options. 2. **Key facts and formulas:** - The area of a polygon is given or can be computed from side lengths and angles. - The disks are congruent and tangent, arranged inside each polygon. - The ratio $X$ (where $X$ is $T$, $R$, or $H$) is defined as $$X = \frac{\text{total area of disks}}{\text{area of polygon}}.$$ - The area of each disk is $\pi r^2$, where $r$ is the radius. 3. **Analyzing each polygon and disk arrangement:** **Equilateral triangle (T):** - Contains 4 mutually tangent congruent disks. - The disks are arranged so that they fit tightly inside the triangle. - The triangle area with side length $a$ is $$A_{\triangle} = \frac{\sqrt{3}}{4}a^2.$$ - The disks' radius $r$ depends on $a$ and the packing arrangement. **Rhombus with 60° angle (R):** - Contains 2 mutually tangent congruent disks arranged vertically. - The rhombus area with side length $a$ and angle $60^\circ$ is $$A_{\text{rhombus}} = a^2 \sin 60^\circ = \frac{\sqrt{3}}{2}a^2.$$ - The disks' radius $r$ depends on $a$ and the packing. **Regular hexagon (H):** - Contains 7 mutually tangent congruent disks: one in the center and six surrounding it. - The hexagon area with side length $a$ is $$A_{\text{hex}} = \frac{3\sqrt{3}}{2}a^2.$$ - The disks' radius $r$ is related to $a$ by the hexagon's geometry. 4. **Known geometric packing results:** - The hexagon packing with 7 disks is a classic close packing with disks tangent to each other and the hexagon. - The rhombus with 2 disks is simpler, and the triangle with 4 disks is more constrained. 5. **Comparing ratios:** - The hexagon packing is the densest among these shapes for the given number of disks. - The rhombus and triangle have similar packing densities for these disk counts. 6. **Conclusion from geometric reasoning and known results:** - $H < R = T$ is true. - This matches option (B). **Final answer:** (B) $H < R = T$