Solved on Jan 22, 2024

Find the side length of a square with a diagonal of $8\mathrm{~cm}$ .

STEP 1

Assumptions
1. The figure is a square.
2. The length of the diagonal is given as $8 \text{ cm}$ .
3. The relationship between the sides of a square and its diagonal is given by the Pythagorean theorem.

STEP 2

Let's denote the length of one side of the square as $s$ . Since all sides of a square are equal, the square has four sides of length $s$ .

STEP 3

According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
$c^2 = a^2 + b^2$
Where $c$ is the hypotenuse and $a$ and $b$ are the other two sides.

STEP 4

In the case of a square, the diagonal acts as the hypotenuse of a right-angled triangle formed by two adjacent sides of the square.

STEP 5

Therefore, we can apply the Pythagorean theorem to the triangle formed by two sides of the square and the diagonal:
$Diagonal^2 = Side^2 + Side^2$

STEP 6

Since both sides are equal in length, we can simplify the equation:
$Diagonal^2 = 2 \times Side^2$

STEP 7

Now, plug in the given value for the diagonal to find the length of the side:
$8^2 = 2 \times Side^2$

STEP 8

Calculate the square of the diagonal:
$64 = 2 \times Side^2$

STEP 9

Divide both sides by 2 to isolate $Side^2$ :
$\frac{64}{2} = Side^2$

STEP 10

Calculate the result:
$32 = Side^2$

STEP 11

To find the length of the side, we take the square root of both sides:
$Side = \sqrt{32}$

STEP 12

Simplify the square root:
$Side = \sqrt{16 \times 2}$

STEP 13

Since the square root of 16 is 4, we have:
$Side = 4 \sqrt{2}$
The length of the sides of the square is $4 \sqrt{2} \text{ cm}$ .

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Find the side length of a square with a diagonal of $8\mathrm{~cm}$ .

STEP 1

Assumptions
1. The figure is a square.
2. The length of the diagonal is given as $8 \text{ cm}$ .
3. The relationship between the sides of a square and its diagonal is given by the Pythagorean theorem.

STEP 2

Let's denote the length of one side of the square as $s$ . Since all sides of a square are equal, the square has four sides of length $s$ .

STEP 3

According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
$c^2 = a^2 + b^2$
Where $c$ is the hypotenuse and $a$ and $b$ are the other two sides.

STEP 4

In the case of a square, the diagonal acts as the hypotenuse of a right-angled triangle formed by two adjacent sides of the square.

STEP 5

Therefore, we can apply the Pythagorean theorem to the triangle formed by two sides of the square and the diagonal:
$Diagonal^2 = Side^2 + Side^2$

STEP 6

Since both sides are equal in length, we can simplify the equation:
$Diagonal^2 = 2 \times Side^2$

STEP 7

Now, plug in the given value for the diagonal to find the length of the side:
$8^2 = 2 \times Side^2$

STEP 8

Calculate the square of the diagonal:
$64 = 2 \times Side^2$

STEP 9

Divide both sides by 2 to isolate $Side^2$ :
$\frac{64}{2} = Side^2$

STEP 10

Calculate the result:
$32 = Side^2$

STEP 11

To find the length of the side, we take the square root of both sides:
$Side = \sqrt{32}$

STEP 12

Simplify the square root:
$Side = \sqrt{16 \times 2}$

STEP 13

Since the square root of 16 is 4, we have:
$Side = 4 \sqrt{2}$
The length of the sides of the square is $4 \sqrt{2} \text{ cm}$ .

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Find the side length of a square with a diagonal of 8 cm8\mathrm{~cm}8 cm.

STEP 1

Assumptions1. The figure is a square.2. The length of the diagonal is given as 8 cm8 \text{ cm}8 cm.3. The relationship between the sides of a square and its diagonal is given by the Pythagorean theorem.

STEP 2

Let's denote the length of one side of the square as sss. Since all sides of a square are equal, the square has four sides of length sss.

STEP 3

According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.c2=a2+b2c^2 = a^2 + b^2c2=a2+b2Where ccc is the hypotenuse and aaa and bbb are the other two sides.

STEP 4

In the case of a square, the diagonal acts as the hypotenuse of a right-angled triangle formed by two adjacent sides of the square.

STEP 5

Therefore, we can apply the Pythagorean theorem to the triangle formed by two sides of the square and the diagonal:Diagonal2=Side2+Side2Diagonal^2 = Side^2 + Side^2Diagonal2=Side2+Side2

STEP 6

Since both sides are equal in length, we can simplify the equation:Diagonal2=2×Side2Diagonal^2 = 2 \times Side^2Diagonal2=2×Side2

STEP 7

Now, plug in the given value for the diagonal to find the length of the side:82=2×Side28^2 = 2 \times Side^282=2×Side2

STEP 8

Calculate the square of the diagonal:64=2×Side264 = 2 \times Side^264=2×Side2

STEP 9

Divide both sides by 2 to isolate Side2Side^2Side2:642=Side2\frac{64}{2} = Side^2264​=Side2

STEP 10

Calculate the result:32=Side232 = Side^232=Side2

STEP 11

To find the length of the side, we take the square root of both sides:Side=32Side = \sqrt{32}Side=32​

STEP 12

Simplify the square root:Side=16×2Side = \sqrt{16 \times 2}Side=16×2​

STEP 13

Since the square root of 16 is 4, we have:Side=42Side = 4 \sqrt{2}Side=42​The length of the sides of the square is 42 cm4 \sqrt{2} \text{ cm}42​ cm.

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Find the side length of a square with a diagonal of $8\mathrm{~cm}$ .

Assumptions
1. The figure is a square.
2. The length of the diagonal is given as $8 \text{ cm}$ .
3. The relationship between the sides of a square and its diagonal is given by the Pythagorean theorem.

Let's denote the length of one side of the square as $s$ . Since all sides of a square are equal, the square has four sides of length $s$ .

According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
$c^2 = a^2 + b^2$
Where $c$ is the hypotenuse and $a$ and $b$ are the other two sides.

Therefore, we can apply the Pythagorean theorem to the triangle formed by two sides of the square and the diagonal:
$Diagonal^2 = Side^2 + Side^2$

Since both sides are equal in length, we can simplify the equation:
$Diagonal^2 = 2 \times Side^2$

Now, plug in the given value for the diagonal to find the length of the side:
$8^2 = 2 \times Side^2$

Calculate the square of the diagonal:
$64 = 2 \times Side^2$

Divide both sides by 2 to isolate $Side^2$ :
$\frac{64}{2} = Side^2$

Calculate the result:
$32 = Side^2$

To find the length of the side, we take the square root of both sides:
$Side = \sqrt{32}$

Simplify the square root:
$Side = \sqrt{16 \times 2}$

Since the square root of 16 is 4, we have:
$Side = 4 \sqrt{2}$
The length of the sides of the square is $4 \sqrt{2} \text{ cm}$ .