A square with a side length of 2s has a smaller square inscribed. The vertices of the smaller square are at the midpoints of the sides of the larger square. What is the ratio of the larger square to the area of the smaller square? Express your answer in simplest form.
Well, let me just size this situation up for you. It sounds like we've got a square with side length 2s and a smaller square hanging out inside it.
Now, if the vertices of the smaller square are at the midpoints of the sides of the larger square, we know that the side length of the smaller square is just s, right? Because it's halfway between the vertices of the larger square.
So, let's calculate the areas of these squares. The area of the larger square is given by (2s)^2, which simplifies to 4s^2. And the area of the smaller square is just s^2.
Now, all we need to do is divide the area of the larger square by the area of the smaller square to find the ratio. So, we have (4s^2)/(s^2). If we simplify that, we get 4.
Therefore, the ratio of the larger square to the area of the smaller square is 4:1, or simply 4.
To solve this question, we need to find the ratio of the area of the larger square to the area of the smaller square.
Let's start by finding the area of the larger square. The formula to find the area of a square is A = s^2, where s is the length of the side.
In this case, the side length of the larger square is 2s. So the area of the larger square is A_larger = (2s)^2 = 4s^2.
Now, let's find the area of the smaller square. Since the vertices of the smaller square are at the midpoints of the larger square, the side length of the smaller square is half the side length of the larger square. So the side length of the smaller square is s.
Therefore, the area of the smaller square is A_smaller = s^2.
Finally, let's find the ratio of the larger square to the smaller square by dividing their areas:
Ratio = A_larger / A_smaller = (4s^2) / (s^2) = 4/1 = 4.
So, the ratio of the larger square to the area of the smaller square is 4:1.
Answer: The ratio is 4:1.
To find the ratio of the larger square to the area of the smaller square, we need to determine the side length of the smaller square in terms of the side length of the larger square.
Let's consider one side of the larger square. Since the vertices of the smaller square are at the midpoints of the sides of the larger square, each side of the smaller square is half the length of the corresponding side of the larger square.
So, the side length of the smaller square is s/2.
The area of the larger square is found by squaring its side length: (2s)^2 = 4s^2.
The area of the smaller square is found by squaring its side length: (s/2)^2 = s^2/4.
Now, we can calculate the ratio of the larger square to the area of the smaller square:
(4s^2)/(s^2/4) = 4s^2 * 4/s^2 = 16.
Therefore, the ratio of the larger square to the area of the smaller square is 16:1.
Did you make your sketch?
area of larger square = (2s)^2 = 4s^2
For the side of the smaller square:
x^2 = s^2 + s^2
x = √2s
area of smaller is (√2s)^2 = 2s^2
so the ratio of their areas = 4s^2 : 2s^2 = 2 : 1