How do you construct a square WXYZ given diagonal WY of length 8 cm and measure its side?

Draw a line segment $\overline{WY}$ of length $8 \text{ cm}$. This will be one of the diagonals of the square.

Construct the perpendicular bisector of $\overline{WY}$.

• With W as the center and a radius greater than $4 \text{ cm}$ (half of $8 \text{ cm}$), draw arcs above and below $\overline{WY}$.
• With Y as the center and the same radius, draw arcs that intersect the previously drawn arcs.
• Draw a straight line connecting the two intersection points. This line is the perpendicular bisector of $\overline{WY}$. Let the point where it intersects $\overline{WY}$ be O.

The diagonals of a square bisect each other at $90^\circ$. So, O is the center of the square, and the other diagonal $\overline{XZ}$ will lie along the perpendicular bisector.

• With O as the center and a radius of $4 \text{ cm}$ (half of the diagonal length), draw arcs on the perpendicular bisector. Mark the points where these arcs intersect the bisector as X and Z.

Connect the points W, X, Y, and Z in order to form the square WXYZ.

In a square, if $s$ is the side length and $d$ is the diagonal length, they are related by the Pythagorean theorem: $s^2 + s^2 = d^2$.

Given the diagonal $d = 8 \text{ cm}$:

To rationalize the denominator:

Calculating the numerical value:

The length of one side of the square is approximately $\text{5.66 cm}$.