Chapter 6.5, Problem 41PPS

To determine

The coordinate proof of each statement.

Explanation of Solution

Given:

The given statement is:

The diagonals of a square are perpendicular.

Calculation:

Place the rectangle named ABCD in the I quadrant.

• Locate point A at origin.
• Locate point B at $\left(a,0\right),\text{ where a = AB}\text{.}$
• Locate point D at $\left(0,a\right),\text{ where AD = a}\text{.}$
• Locate point C at $\left(a,a\right),\text{ such that, CD = a and }\overline{\text{DC}}$ is horizontal.

The figure will be:

Using the slope formula to check whether the diagonals are congruent;

The slope of AC will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{a-0}{a-0}\\ m=\frac{a}{a}\\ m=1\end{array}$

The slope of BD will be:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{a-0}{0-a}\\ m=\frac{a}{-a}\\ m=-1\end{array}$

Hence,

  $\begin{array}{l}as,1\left(-1\right)=-1\\ \therefore \overline{AC}\perp \overline{BD}\end{array}$