91Ó°ÊÓ

In this exercise, we need to prove that the diagonals of a rectangle are congruent using analytic geometry. Diagonal congruence means that the two diagonals of a rectangle have equal lengths. To establish this proof, we will use the properties of a rectangle and the distance formula. Consider a rectangle with vertices placed on a coordinate grid, such as A \(0,0\), B \(a,0\), C \(a,b\), and D \(0,b\). The diagonals in question are AC and BD. Our aim will be to show these two diagonals have the same length.

The distance formula helps us find the length between two points in a plane. This is the backbone of our proof for the congruence of diagonals in a rectangle. The formula is derived from the Pythagorean theorem and is given by \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. To prove the congruence of diagonals, we will use this formula to calculate the length of each diagonal.
For diagonal AC, the coordinates of A are \(0, 0\) and C are \(a, b\). So, applying the distance formula, we get \[ AC = \sqrt{ (a-0)^2 + (b-0)^2 } = \sqrt{ a^2 + b^2 } \].
Similarly, for diagonal BD, the coordinates of B are \(a,0\) and D are \(0,b\). Applying the distance formula here, we get \[ BD = \sqrt{ (a-0)^2 + (0-b)^2 } = \sqrt{ a^2 + b^2 } \].
Since both lengths are equal to \( \sqrt{ a^2 + b^2 } \), we have successfully shown that the diagonals AC and BD are congruent.

Understanding the placement of rectangle vertices on a coordinate plane is key to applying the distance formula accurately. Let's consider our rectangle ABCD again. When defining rectangle vertices:

• A is at the origin (0,0)
• B is at point (a,0) on the x-axis
• C is at point (a,b) on the coordinate plane
• D is at point (0,b) on the y-axis

By fixing these points' positions, we simplify the calculations. The lengths of sides AB and AD are equal to the lengths along the x-axis and y-axis, respectively. This systematic arrangement ensures the diagonals intersect at right angles, forming two equal right triangles, thus making the proof and calculations straightforward. The coordinate grid lets us directly apply the distance formula to find the lengths of the diagonals AC and BD, ultimately verifying that they are congruent.