The sides of a quadrilateral are 3, 4, 5, and 6. Find the length of the shortest side of a similar quadrilateral whose area is 9 times as great.

9,10,11,12

To solve this problem, we need to understand the concept of similarity between two figures.

Similar figures have corresponding sides that are proportional, meaning their ratios are equal. In the case of quadrilaterals, this means that corresponding sides of the two quadrilaterals will be in the same ratio.

Let's assume that the shortest side of the original quadrilateral (with sides 3, 4, 5, and 6) is 'x'.

To find the length of the shortest side of the similar quadrilateral, we need to determine the scale factor between the two quadrilaterals' corresponding sides.

The scale factor (k) is defined as the ratio of the corresponding sides between the similar quadrilaterals. In this case, we are given that the area of the similar quadrilateral is 9 times as great as the original, so the scale factor for area (K_area) is the square root of 9, which equals 3.

Now, since the corresponding sides of the two quadrilaterals are in the same ratio, we can set up a proportion:

x / 3 = 3 / 6

To solve for x, cross-multiply:

6x = 9 * 3

6x = 27

Now, divide both sides by 6 to solve for x:

x = 27 / 6

Simplifying, we find that x is equal to 4.5.

Therefore, the length of the shortest side of the similar quadrilateral is 4.5.