Answer :
To solve this question, we need to understand how the perimeter of a square is related to the length of its sides and how we can use this information to find the diagonal length.
Step 1: Find the length of one side of the square.
The perimeter (P) of a square is the sum of all four sides. Since all sides of a square are equal, if the perimeter is 56, we can find the length of one side (S) using the formula:
\[ P = 4 \cdot S \]
So, to find the length of one side, we divide the perimeter by 4:
\[ S = \frac{P}{4} = \frac{56}{4} = 14 \]
Step 2: Find the length of the diagonal of the square.
A square's diagonal creates two right triangles when we draw it from one corner to the opposite corner, splitting the square exactly in half. We can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b):
\[ c^2 = a^2 + b^2 \]
In the case of the square, both a and b are equal to the side length of the square, S, and c is the length of the diagonal (D). Therefore, the equation becomes:
\[ D^2 = S^2 + S^2 \]
\[ D^2 = 2 \cdot S^2 \]
Substitute the value of S (14) into the equation to find the D:
\[ D^2 = 2 \cdot (14)^2 \]
\[ D^2 = 2 \cdot 196 \]
\[ D^2 = 392 \]
Now take the square root of both sides to solve for D:
\[ D = \sqrt{392} \]
To simplify the expression further, we look for perfect squares in the prime factorization of 392, which can be taken out of the square root.
\[ 392 = 2^3 \cdot 7^2 \]
We have a perfect square \( 7^2 \), which we can take outside the square root, and \( 2^3 = 8 \) where \( 2^1 \) can be taken out as \( 2 \), leaving \( 2^1 = 2 \) inside the root. Therefore, the simplest radical form is:
\[ D = 2 \cdot 7 \cdot \sqrt{2} \]
\[ D = 14 \cdot \sqrt{2} \]
The length of the diagonal of the square, in simplest radical form, is \( 14\sqrt{2} \).
Answer:
To find the length of a diagonal of a square, we first need to determine the length of one side of the square. The perimeter ( P ) of a square is equal to four times the length of one side ( s ), so we can write:
[ P = 4s ]
Given that the perimeter ( P ) is 56, we can solve for ( s ):
[ 56 = 4s ] [ s = \frac{56}{4} ] [ s = 14 ]
Now, to find the diagonal ( d ) of the square, we can use the Pythagorean theorem, which in the case of a square with side length ( s ) is:
[ d = s\sqrt{2} ]
Substituting the side length ( s = 14 ) into the equation, we get:
[ d = 14\sqrt{2} ]
Therefore, the length of the diagonal of the square in simplest radical form is ( \mathbf{14\sqrt{2}} ).
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Step-by-step explanation:
