this diagram shows the design for a quilt.
the quilt is made of 4 squares that have beed divided along their diagonals.
the length of bd is 4.
a. find the length of ae.
type your answer in the box.
ae = \boxed{} unit(s)
b. find the area of square abcd.
type your answer in the box.

Question

Accepted Answer

### Part a
# Explanation:
## Step 1: Identify the square and its diagonal
ABCD is a square (since the quilt is made of squares, and BD is a diagonal of square ABCD). In a square, the diagonals are equal and bisect each other. Also, the quilt is made of 4 smaller squares, so E, F, G, H are mid - points. BD is the diagonal of square ABCD with length 4. The side of the smaller square (AE is a side of a smaller square) can be found by considering the relationship between the diagonal of the large square and the side of the small square.

The large square ABCD has diagonal $ BD = 4 $. The length of the side of the large square $ s $ can be found using the Pythagorean theorem for a square: for a square with side $ s $ and diagonal $ d $, $ d=\sqrt{s^{2}+s^{2}}=\sqrt{2s^{2}} = s\sqrt{2} $. But also, the large square is divided into 4 smaller squares. So the side of the large square $ s $ is equal to $ 2\times $ side of the small square (let the side of the small square be $ x $, so $ s = 2x $). The diagonal of the large square $ BD=4 $, and also, the diagonal of the large square is equal to the length of $ 2 $ times the diagonal of the small square? Wait, no. Alternatively, since E is the mid - point of AD, and AD is a side of square ABCD. In square ABCD, the diagonal $ BD = 4 $. The side of square ABCD: let the side of square ABCD be $ a $. Then by Pythagorean theorem in square ABCD, $ BD^{2}=a^{2}+a^{2}$, so $ 16 = 2a^{2}$, $ a^{2}=8 $, $ a = 2\sqrt{2} $. But wait, the quilt is made of 4 squares, so AD is divided into two equal parts by E, so $ AE=\frac{AD}{2} $. Since AD is a side of square ABCD, and we know that in square ABCD, the diagonal $ BD = 4 $. Also, in a square, the diagonal is equal to $ \sqrt{2}\times $ side. But also, the large square is composed of 4 small squares, so the side of the large square is twice the side of the small square. Let the side of the small square be $ x $, then the side of the large square is $ 2x $. The diagonal of the large square $ BD=\sqrt{(2x)^{2}+(2x)^{2}}=\sqrt{8x^{2}}=2x\sqrt{2} $. We know $ BD = 4 $, so $ 2x\sqrt{2}=4 $, $ x\sqrt{2}=2 $, $ x=\frac{2}{\sqrt{2}}=\sqrt{2} $? Wait, no, maybe a simpler way. Since the quilt is made of 4 squares, the diagonal of the large square (BD = 4) is equal to the length of 2 times the diagonal of the small square? No, actually, AE is a side of a small square. The large square has 4 small squares, so the side of the large square is $ 2\times $ side of small square. The diagonal of the large square $ BD = 4 $, and the side of the large square $ s $, so $ s\sqrt{2}=4 $, $ s=\frac{4}{\sqrt{2}} = 2\sqrt{2} $. Then the side of the small square (AE) is $ \frac{s}{2}=\frac{2\sqrt{2}}{2}=\sqrt{2} $? Wait, no, that's wrong. Wait, another approach: The quilt is made of 4 squares, so the diagonals of the small squares are equal to half of the diagonal of the large square? No, BD is the diagonal of the large square. The large square is divided into 4 smaller squares, so the mid - points E, F, G, H divide the sides of the large square into two equal parts. So AE is half of AD, and AD is a side of the large square. In a square, if the diagonal $ d = 4 $, then the side $ s=\frac{d}{\sqrt{2}}=\frac{4}{\sqrt{2}} = 2\sqrt{2} $. Then $ AE=\frac{s}{2}=\frac{2\sqrt{2}}{2}=\sqrt{2} $? Wait, no, maybe I made a mistake. Wait, actually, the large square is composed of 4 small squares, so the length of BD (the diagonal of the large square) is equal to the length of 2 times the side of the large square? No, no. Wait, let's look at the diagram. The points E, F, G, H are mid - points. So AE is a side of a small square, and the large square ABCD has side length equal to $ 2\times AE $. The diagonal of the large square $ BD $ in a square with side $ 2x $ (where $ x = AE $) is $ (2x)\sqrt{2} $. We know $ BD = 4 $, so $ 2x\sqrt{2}=4 $, $ x\sqrt{2}=2 $, $ x=\frac{2}{\sqrt{2}}=\sqrt{2} $. Wait, but maybe there is a simpler way. Since the quilt is made of 4 squares, the diagonal of the large square is equal to 2 times the side of the large square? No, that's not the case. Wait, no, in the diagram, BD is a diagonal, and the large square is divided into 4 small squares, so the length of AE is equal to half of the side of the large square. And the side of the large square can be found using the fact that in a square, if the diagonal is 4, then the side is $ \frac{4}{\sqrt{2}}=2\sqrt{2} $, so half of that is $ \sqrt{2} $. But wait, maybe the large square is actually a square where the diagonal is equal to 4, and the side of the small square is 1? No, that can't be. Wait, maybe I misinterpret the diagram. The quilt is made of 4 squares that have been divided along their diagonals. So ABCD is a square, and it is divided into 4 smaller squares: AHE, EHD, DFC, and FCB? No, maybe ABCD is a square, and E, F, G, H are mid - points, so AE = ED = DH =... Wait, BD is a diagonal of the large square ABCD. The length of BD is 4. In a square, the diagonals are equal and bisect each other at 90 degrees. Also, the 4 small squares: so AE is a side of a small square, and the large square has side length equal to 2 times AE. Let the side of the small square be $ x $, so the side of the large square is $ 2x $. The diagonal of the large square $ BD=\sqrt{(2x)^{2}+(2x)^{2}}=\sqrt{8x^{2}} = 2x\sqrt{2} $. We know $ BD = 4 $, so $ 2x\sqrt{2}=4 $, $ x\sqrt{2}=2 $, $ x=\frac{2}{\sqrt{2}}=\sqrt{2} \approx 1.414 $. But maybe there is a mistake here. Wait, another way: The quilt is made of 4 squares, so the length of BD (the diagonal) is equal to the length of 2 times the side of the large square? No, that's not the Pythagorean theorem. Wait, maybe the large square has side length 4? No, because BD is the diagonal. Wait, I think I made a mistake. Let's start over.

ABCD is a square. BD is a diagonal of square ABCD, so $ BD = 4 $. In a square, if the diagonal is $ d $, the side $ s $ is given by $ s=\frac{d}{\sqrt{2}} $. So $ s=\frac{4}{\sqrt{2}} = 2\sqrt{2} $. Now, E is the mid - point of AD, so $ AE=\frac{AD}{2} $. Since $ AD = s=2\sqrt{2} $, then $ AE=\frac{2\sqrt{2}}{2}=\sqrt{2} \approx 1.414 $. But maybe the problem is that the large square is actually composed of 4 small squares, so the side of the large square is 2 times the side of the small square, and the diagonal of the large square is 4. So if the side of the small square is $ x $, the side of the large square is $ 2x $, and the diagonal of the large square is $ (2x)\sqrt{2}=4 $, so $ x=\frac{4}{2\sqrt{2}}=\frac{2}{\sqrt{2}}=\sqrt{2} $. So $ AE = x=\sqrt{2} $? Wait, no, AE is the side of the small square, so $ AE=\sqrt{2} $. But maybe the answer is 1? No, that doesn't fit. Wait, maybe the large square is a square with side length 4? No, because BD is the diagonal. Wait, I think I have a wrong approach. Let's consider that the quilt is made of 4 squares, so the length of AE is equal to half of the side of the large square, and the diagonal of the large square is 4. In a square, side $ s $, diagonal $ d = s\sqrt{2} $, so $ s=\frac{d}{\sqrt{2}} $. Then $ AE=\frac{s}{2}=\frac{d}{2\sqrt{2}} $. Substituting $ d = 4 $, we get $ AE=\frac{4}{2\sqrt{2}}=\frac{2}{\sqrt{2}}=\sqrt{2} \approx 1.414 $. But maybe the problem is designed so that the large square has a diagonal of 4, and the side of the small square is 1. Wait, no, that would mean the diagonal of the large square is $ 2\sqrt{2}\approx2.828 $, not 4. So my initial calculation seems correct.

## Step 2: Calculate AE
Since AE is the side of a small square, and the large square has diagonal $ BD = 4 $. The side of the large square $ s=\frac{BD}{\sqrt{2}}=\frac{4}{\sqrt{2}} = 2\sqrt{2} $. Since E is the mid - point of AD, $ AE=\frac{s}{2}=\frac{2\sqrt{2}}{2}=\sqrt{2} $. But wait, maybe there is a simpler way. If we consider that the 4 small squares are such that the diagonal of the large square is equal to 2 times the side of the large square? No, that's not the case. Wait, maybe the diagram is such that ABCD is a square with side length 2, so the diagonal is $ 2\sqrt{2}\approx2.828 $, but the problem says BD is 4. I think I made a mistake in the diagram interpretation. Let's re - interpret: The quilt is made of 4 squares that have been divided along their diagonals. So ABCD is a square, and it is divided into 4 smaller squares, each with side length $ x $. Then the side of the large square is $ 2x $, and the diagonal of the large square is $ 2x\sqrt{2} $. We know the diagonal $ 2x\sqrt{2}=4 $, so $ x\sqrt{2}=2 $, $ x = \frac{2}{\sqrt{2}}=\sqrt{2} $. So $ AE=x=\sqrt{2} $.

# Answer:
$ \sqrt{2} $ (or approximately 1.414, but in exact form $ \sqrt{2} $)

### Part b
# Explanation:
## Step 1: Recall the formula for the area of a square
The area of a square with side length $ s $ is $ A = s^{2} $. For a square with diagonal $ d $, the area can also be calculated as $ A=\frac{d^{2}}{2} $ (since $ d = s\sqrt{2}$, so $ s=\frac{d}{\sqrt{2}} $, and $ A = s^{2}=\frac{d^{2}}{2} $).

## Step 2: Calculate the area of square ABCD
We know that the diagonal of square ABCD, $ BD = 4 $. Using the formula for the area of a square in terms of its diagonal $ A=\frac{d^{2}}{2} $, substitute $ d = 4 $ into the formula. Then $ A=\frac{4^{2}}{2}=\frac{16}{2}=8 $.

# Answer:
8

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Part a

Step 1: Identify the square and its diagonal

Step 1: Recall the formula for the area of a square

Step 2: Calculate the area of square ABCD

Answer:

Part b