the area of ABCDE is 18 sq. cm.; the area of a similar pentagon PQRST is 32 sq.cm. the diagonal AC is 6 cm.; find the length of PR.

Assumptions:

1. ABCDE is also a pentagon
2. PR is the corresponding diagonal in the 2nd pentagon

In that case, the areas of similar figures are proportional to the square of their sides (or their diagonals)

PR^2/6^2 = 18/32
PR^2 = 648/32 = 81/4
PR = 9/2

To find the length of PR in the similar pentagon PQRST, we can set up a proportion using the areas of the two pentagons.

The given area of ABCDE is 18 sq. cm, and the area of PQRST is 32 sq. cm. Since the two pentagons are similar, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.

Let's say the length of AC in ABCDE is x cm. Based on the given information, we know that the diagonal AC is 6 cm, so x = 6 cm.

Now, let's set up a proportion to find the length of PR. Let y be the length of PR.

(Length of PR)^2 / (Length of AC)^2 = Area of PQRST / Area of ABCDE

y^2 / 6^2 = 32/18

y^2 / 36 = 32/18

Cross-multiplying, we get:

18y^2 = 32 * 36

Dividing both sides of the equation by 18:

y^2 = (32 * 36) / 18

y^2 = 64

Taking the square root of both sides:

y = √64

y = 8 cm

Therefore, the length of PR in the similar pentagon PQRST is 8 cm.