(a) Let the scale factor of the perimeter from Figure A to Figure B be denoted by k. The ratio of the perimeters of two similar figures is equal to the scale factor of the corresponding lengths.
Since the area of Figure A is 36 square meters and its side length is 3 meters, we can calculate its perimeter as:
Perimeter of Figure A = 4 × 3 = 12 meters
Similarly, since the area of Figure B is 81 square meters, we can calculate its side length as:
Side length of Figure B = √(81) = 9 meters
Therefore, the perimeter of Figure B is:
Perimeter of Figure B = 4 × 9 = 36 meters
Now we can use the ratio of the perimeters to find the scale factor:
k = Perimeter of Figure B / Perimeter of Figure A
k = 36 / 12
k = 3
Therefore, the scale factor of the perimeter from Figure A to Figure B is 3.
(b) Since the scale factor of the lengths from Figure A to Figure B is also 3 (because the area scale factor is the square of the length scale factor), we can find the length of the corresponding side in Figure B by multiplying the length of the corresponding side in Figure A by 3.
If the side length of Figure A is 3 meters, then the length of the corresponding side in Figure B is:
Length of corresponding side in Figure B = 3 × 3
Length of corresponding side in Figure B = 9 meters
Therefore, the length of the corresponding side in Figure B is 9 meters.