The diagonal of a square is sqrt 84. What is the area of the square.

Expert Answers
baxthum8 eNotes educator| Certified Educator

The diagonal of a square is the hypotenuse of a 45-45-90 triangle.

In a 45-45-90 triangle the hypotenuse is the leg multiplied by `sqrt(2).`

Therefore, if given the hypotenuse, divide by `sqrt(2)`  to find the length of the leg.

`sqrt(84)/sqrt(2) =sqrt(42)`

Therefore each side of the square is `sqrt(42).`

To find the area of the square:  `A=s^2.`

`A = (sqrt(42))^2`

Therefore the area of the square is: `42 u^2`

justaguide eNotes educator| Certified Educator

The length of the diagonal of a square with side S is equal to `sqrt(S^2 + S^2)` = `S*sqrt 2` .

If the diagonal of a square is D, the length of the side is `D/sqrt 2`

A square has a diagonal of length `sqrt 84` . The area of the square is `(sqrt 84)^2/2 = 84/2 = 42`

The area of the square is 42 square units.

tonys538 | Student

Another way to look at this is to use the formula for the area of a rhombus. A rhombus is a quadrilateral with 4 equal sides. If the length of the diagonals is D1 and D2, the area of the rhombus is `(D1*D2)/2` .

A square is a rhombus and as the adjacent sides are perpendicular to each other, the diagonals are equal in length.

The area of a square with diagonal `sqrt 84` is : `(sqrt 84*sqrt 84)/2 = 84/2 = 41`

The area of the square is 41.