Question

Let ( ABCD ) be a square. Let ( E, F, G, ) and ( H ) be the centers, respectively, of equilateral triangles with bases ( AB, BC, CD, ) and ( DA ), each exterior to the square. What is the ratio of the area of square ( EFGH ) to the area of square ( ABCD )?

a) ( 1:1 )
b) ( 2:1 )
c) ( 3:1 )
d) ( 4:1 )

Answer 1

The ratio of the area of square EFGH to the area of square ABCD is 1:1.

Step-by-step explanation:

To find the ratio of the area of square EFGH to the area of square ABCD, we need to determine the relationship between the side lengths of the two squares. Let's assume the side length of square ABCD is 's'.

Since E, F, G, and H are the centers of equilateral triangles with bases AB, BC, CD, and DA, respectively, the side lengths of these triangles will be equal to the side length of the square ABCD. Therefore, the side length of each equilateral triangle will be 's'.

The ratio of the area of square EFGH to the area of square ABCD can be calculated as follows:

(Area of square EFGH) / (Area of square ABCD) = [(side length of square EFGH)^2] / [(side length of square ABCD)^2] = (s^2) / (s^2) = 1.

Hence, the ratio of the area of square EFGH to the area of square ABCD is 1:1.