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.
A. 9
B. 13.5
C. 27
I would say C but I'm not that far in Geometry
area ratio = length ratio^2
if area ratio is 9
length ratio is 3
3 * 3 = 9 is the shortest side of the big one
To find the length of the shortest side of a similar quadrilateral whose area is 9 times as great as the given quadrilateral, we can use the concept of similarity.
Let's assume the length of the shortest side of the similar quadrilateral is x.
The ratio of the areas of similar figures is equal to the square of the ratio of their corresponding sides. Therefore, we can set up the following equation:
(x/3)^2 = 9
Simplifying the equation, we have:
(x^2/9) = 9
Multiplying both sides by 9, we get:
x^2 = 81
Taking the square root of both sides, we find:
x = 9
Therefore, the length of the shortest side of the similar quadrilateral is 9.
The answer is A. 9.
To find the length of the shortest side of a similar quadrilateral whose area is 9 times as great, we need to understand the relationship between the lengths and areas of similar figures.
In a plane figure, if the lengths of corresponding sides are scaled by a factor of k, then the areas of the two figures are scaled by a factor of k^2.
Let's say the length of the shortest side of the original quadrilateral is x. We know that the area of the original quadrilateral is A, and the area of the similar quadrilateral is 9A.
The relationship between the areas can be expressed as (9A) = (k^2)(A), where k is the scaling factor for the sides. Simplifying this equation gives us 9 = k^2.
Since we are interested in the length of the shortest side of the similar quadrilateral, we need to find the scaling factor k that satisfies k^2 = 9. Taking the square root of both sides gives us k = 3.
Therefore, the scaling factor for the sides is 3. To find the length of the shortest side of the similar quadrilateral, we multiply the length of the corresponding side in the original quadrilateral (x) by the scaling factor (3): shortest side of similar quadrilateral = 3x.
Now we can substitute the given lengths of the original quadrilateral's sides (3, 4, 5, and 6) and solve for x.
We have the following equations:
3x = 3,
4x = 4,
5x = 5,
6x = 6.
When we solve these equations, we find that x = 1.
Finally, we can calculate the length of the shortest side of the similar quadrilateral by multiplying x by the scaling factor: shortest side of similar quadrilateral = 3x = 3 * 1 = 3.
Thus, the length of the shortest side of the similar quadrilateral whose area is 9 times as great is 3.
Therefore, the correct answer is not provided in the options.