Understanding the Diagonal of a Square
The diagonal of a square is a line segment connecting two non-adjacent vertices (opposite corners). Every square has two diagonals that are equal in length and bisect each other at right angles (90 degrees).
Diagonal Formulas
Diagonal from Side
Apply the Pythagorean theorem to the right triangle formed by two sides and the diagonal.
d = s√2
Side from Diagonal
Rearrange the diagonal formula to solve for the side length.
s = d / √2 = d√2 / 2
Derivation Using the Pythagorean Theorem
In a square with side length s, the diagonal forms a right triangle with two sides of the square. By the Pythagorean theorem:
d² = s² + s² = 2s², therefore d = s√2.
Properties of Square Diagonals
- Both diagonals are equal in length.
- They bisect each other at 90-degree angles.
- Each diagonal divides the square into two congruent right isosceles triangles.
- The diagonals bisect the vertex angles into 45-degree angles.
- The intersection point of the diagonals is the center of the square.