Square EFGH has one vertex on each side of square ABCD. Point E is on AB with AE = 7.EB . What is the ratio of the area of EFGH to the area of ABCD ?

To solve the problem, we need to find the ratio of the area of square EFGH to the area of square ABCD, given that point E is on side AB of square ABCD with the condition that AE = 7 * EB. Let's break this down step by step.

Understanding the Squares

Let’s denote the side length of square ABCD as s. The area of square ABCD is then:

Area of ABCD = s²

Locating Point E

Since point E is on side AB, we can express the lengths AE and EB in terms of the side length s:

• Let EB = x.
• Then, AE = 7 * EB = 7x.

Since E lies on AB, we have:

AE + EB = s

Substituting the expressions for AE and EB gives:

7x + x = s

8x = s

From this, we can find x:

x = s/8

Thus, we have:

• EB = s/8
• AE = 7s/8

Finding Other Vertices of Square EFGH

Now, we need to determine the positions of the other vertices F, G, and H. Since E is located at (0, 7s/8) on side AB, we can find the coordinates of the other vertices based on the properties of square EFGH:

• Point F will be on side BC, which is vertical. Since E is at (0, 7s/8), we can place F at (s/8, 7s/8).
• Point G will be on side CD, which is horizontal. We can place G at (s/8, s/8).
• Point H will be on side DA, which is vertical. We can place H at (0, s/8).

Calculating the Area of Square EFGH

To find the area of square EFGH, we need to determine the length of one of its sides. The distance between points E and F gives us the side length:

Distance EF = √[(s/8 - 0)² + (7s/8 - 7s/8)²] = s/8.

Thus, the area of square EFGH is:

Area of EFGH = (s/8)² = s²/64.

Finding the Ratio of Areas

Now we can find the ratio of the area of square EFGH to the area of square ABCD:

Ratio = Area of EFGH / Area of ABCD = (s²/64) / (s²) = 1/64.

Final Result

Therefore, the ratio of the area of square EFGH to the area of square ABCD is:

1:64.