1. The problem asks for the ratio between the areas of two similar polygons given the ratio of their perimeters is 4 : 9. 2. For similar polygons, the ratio of their areas is the square of the ratio of their corresponding sides or perimeters. 3. Given perimeter ratio = 4 : 9, area ratio = $4^2 : 9^2 = 16 : 81$. 4. So, the correct answer is (d) 16 : 81. --- 1. Given $\triangle ABC \sim \triangle XYZ$ and $AB = 3 \times XY$, find $\frac{a(\triangle XYZ)}{a(\triangle ABC)}$. 2. The ratio of corresponding sides is $AB : XY = 3 : 1$. 3. Area ratio is the square of side ratio, so $\frac{a(\triangle ABC)}{a(\triangle XYZ)} = 3^2 : 1^2 = 9 : 1$. 4. Therefore, $\frac{a(\triangle XYZ)}{a(\triangle ABC)} = \frac{1}{9}$. 5. The correct answer is (d) 1/9. --- 1. Given the ratio of areas of two similar polygons is 9 : 49, find the ratio of their corresponding sides. 2. The ratio of areas = square of ratio of sides. 3. So, side ratio = $\sqrt{9} : \sqrt{49} = 3 : 7$. 4. The correct answer is (a) 3 : 7. --- 1. Given lengths of two corresponding sides are 7 cm and 11 cm, find the ratio of their perimeters. 2. The ratio of perimeters is the same as the ratio of corresponding sides. 3. So, perimeter ratio = $7 : 11$. 4. The correct answer is (c) 7/11. --- 1. Given the ratio of corresponding sides of two similar triangles is 2 : 5 and area of the first triangle is 16 cm², find the area of the second triangle. 2. Area ratio = square of side ratio = $2^2 : 5^2 = 4 : 25$. 3. Let area of second triangle be $x$, then $\frac{16}{x} = \frac{4}{25}$. 4. Solving for $x$, $x = \frac{16 \times 25}{4} = 100$ cm². 5. The correct answer is (c) 100. --- 1. Given lengths of corresponding sides are 12 cm and 16 cm, and area of smaller polygon is 135 cm², find the area of the greater polygon. 2. Side ratio = $12 : 16 = 3 : 4$. 3. Area ratio = square of side ratio = $3^2 : 4^2 = 9 : 16$. 4. Let area of greater polygon be $x$, then $\frac{135}{x} = \frac{9}{16}$. 5. Solving for $x$, $x = \frac{135 \times 16}{9} = 240$ cm². 6. The correct answer is (c) 240.