The sides of a quadrilateral are 3, 4, 5, and 6. What is the length of the shortest side of a similar quadrilateral whose area is 9 times as great?
The area of similar figures is proportional to the square of their sides.
So let the shortest side of the second quad be x
x^2/3^2 = 9A/A
x^2/9 = 9/1
x^2 = 81
x = 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 sides and the area of similar quadrilaterals.
In similar figures, corresponding sides are proportional. This means that if we scale the sides of a given quadrilateral by a factor of k, then the corresponding sides of a similar quadrilateral will be multiplied by the same factor k.
Let's denote the length of the shortest side of the original quadrilateral as x. We know that the area of the original quadrilateral is A.
According to the given information, the sides of the original quadrilateral are 3, 4, 5, and 6. To find the area of the original quadrilateral, we can use Heron's formula, which states that the area of a quadrilateral with sides a, b, c, and d is given by:
A = √[s(s - a)(s - b)(s - c)(s - d)]
where s is the semiperimeter given by:
s = (a + b + c + d) / 2.
In our case, the sides of the original quadrilateral are 3, 4, 5, and 6. Therefore, the semiperimeter s is:
s = (3 + 4 + 5 + 6) / 2 = 9 / 2 = 4.5.
Using Heron's formula, we can calculate the area A of the original quadrilateral:
A = √[4.5(4.5 - 3)(4.5 - 4)(4.5 - 5)(4.5 - 6)]
A = √[4.5 * 1.5 * 0.5 * (-0.5) * (-1.5)]
A = √[0.5625]
A ≈ 0.7500.
Now, we want to find the length of the shortest side of a similar quadrilateral whose area is 9 times as great. Let's denote the length of the shortest side of this similar quadrilateral as y. The area of this similar quadrilateral is 9A, which means:
9A = √[s'(s' - y)(s' - b')(s' - c')(s' - d')],
where s' is the semiperimeter of the similar quadrilateral.
Since the sides of a similar quadrilateral are proportional, we can write:
y / x = 9^(1/4),
where 9^(1/4) is the fourth root of 9. Solving this equation for y, we get:
y = x * 9^(1/4).
Now, substituting the length of the shortest side of the original quadrilateral (x ≈ 3) into the equation, we can find the length of the shortest side of the similar quadrilateral whose area is 9 times as great:
y = 3 * 9^(1/4)
y ≈ 3 * 1.6818
y ≈ 5.0454.
Therefore, the length of the shortest side of the similar quadrilateral whose area is 9 times as great is approximately 5.0454.