in a square the lengths of the diagonals are the same. is a square has an area of 32 square feet, then what is the length of each diagonal

Question

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Answer 1

A square has sides of equal length all around.  Area is length times width.

 

Let x = side length

 

Area = x²

 

32 = x²

 

 

Square-root both sides of the equation.

 

√32 = x

 

√16√2 = x

 

4√2 = x

 

 

The diagonals form a right triangle with the sides of the square.  To find the diagonal, we can use Pythagorean theorem.

 

c = √(x² + x²)

 

 

Reduce this formula.

 

c = √(2x²)

 

c = √2 * √(x²)

 

c = x√2

 

 

Now substitute the value of x that we previously found out.

 

c = (4√2) * √2

 

c = 4 * 2

 

c = 8

 

 

Each diagonal is 8 feet.

Answer 2

Hi Frank,

 

Let's first find the length of a side (s) of the square:

    s² = 32

    s = √32

 

Now the length of a diagonal of the square can be found using the side ratios for the sides of a 45-45-90 right triangle:

    1:1:√2

    sides are 1:1 and diagonal is s√2

 

Plugging in our value for s:

   d = s√2

   d = (√32)(√2)

   d = (√16)(√2)(√2)

   d = 4(√2)(√2)

   d = 4(2)

   d = 8

 

We could have also used the Pythagorean Theorem:

   d² = s² + s²

   d² = 32 + 32

   d² = 64

   d = √64

   d = 8

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.