Use coordinate geometry to prove each statement. First draw...

Use coordinate geometry to prove each statement. First draw a figure and choose convenient axes and coordinates.
The diagonals of a rhombus are perpendicular. (Theorem $5-13$ ) (Hint: Let the vertices be $(0,0),(a, 0),(a+b, c),$ and $(b, c)$ . Show that $c^{2}=a^{2}-b^{2} . )$

Key Concepts

Coordinate Geometry

Coordinate geometry involves representing geometric figures on a coordinate plane and using algebraic methods to analyze their properties. In proofs, choosing a convenient coordinate system can simplify the process of establishing relationships between geometric elements, such as verifying distances, slopes, and intersections.

Rhombus

A rhombus is a type of quadrilateral in which all sides are congruent. Its special properties include the fact that its diagonals, while not necessarily equal in length, not only bisect each other but are also perpendicular. These inherent characteristics make the rhombus an ideal candidate for proofs involving symmetry and intersecting lines in the coordinate plane.

Diagonals and Their Perpendicularity

The proof that the diagonals of a rhombus are perpendicular leverages both the geometric properties of the shape and the algebraic properties of slopes. By expressing the vertices of the rhombus in terms of coordinates, one can calculate the slopes of the diagonals and demonstrate that their slopes are negative reciprocals, which is a necessary and sufficient condition for perpendicularity.

Slope and Perpendicular Lines

A fundamental concept in coordinate geometry is the relationship between the slopes of perpendicular lines. Two lines are perpendicular if and only if the product of their slopes is -1. This principle is crucial when using coordinate methods to establish the perpendicularity of diagonals in various quadrilaterals, including the rhombus.

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