The perimeterof a square is 32 inches. Find the lenght of its diagonal. the answers should be in simplest form. make a sketch and show work.
Each side of the square is equal to 8 inches (32/4 = 8).
The diagonal through the square will form 2 right triangles.
Use the Pythagorean Theorem to solve for the length of the diagonal.
a^2 + b^2 = c^2
8^2 + 8^2 = c^2
64 + 64 = c^2
128 = c^2
√128 = c^2
11.3137 is the approximate value of c.
I hope this helps.
√128 = c^2 should read:
√128 = c
Sorry for any confusion
To find the length of the diagonal of a square, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (in this case, the diagonal) is equal to the sum of the squares of the lengths of the other two sides.
Let's start by finding the length of one side of the square. The formula for the perimeter of a square is P = 4s, where P is the perimeter and s is the length of one side. Given that the perimeter is 32 inches, we can set up the equation as follows:
32 = 4s
Dividing both sides by 4, we get:
s = 8 inches
Now, let's find the length of the diagonal. Let's label the square with corners A, B, C, and D, going clockwise. We can draw a diagonal from A to C.
Using the Pythagorean theorem, we have:
AC^2 = AB^2 + BC^2
Since it is a square, AB = BC = s = 8 inches. Substituting these values into the equation, we have:
AC^2 = 8^2 + 8^2
AC^2 = 64 + 64
AC^2 = 128
To find AC, we need to find the square root of 128:
AC = √128
To simplify, we can write 128 as the product of a perfect square and a prime number:
128 = 2^7
Taking the square root of both sides, we have:
AC = √(2^7)
Since the square root (√) and square (^2) are inverse operations, we can simplify:
AC = 2^(7/2)
To simplify further, we can write 2^(7/2) as (2^7)^(1/2):
AC = (2^7)^(1/2)
Using the property of exponents, we multiply the exponents:
AC = 2^(7/2) = (2^(1/2))^7 = √2^7 = √128
Therefore, the length of the diagonal of the square is √128 inches.
To find the simplified form, we can simplify the square root of 128. We can write 128 as the product of a perfect square and a prime number:
128 = 2^7
Thus, the simplified form of the length of the diagonal is 8√2 inches.
I hope this explanation helps! Let me know if you have any further questions.