The coordinates of three vertices of a rectangle are given....

The coordinates of three vertices of a rectangle are given. Plot the points and find the coordinates of the fourth vertex. Is the rectangle a square?
$$O(0,0), P(0,5), Q(\underline{?}, \underline{?}), R(2,0)$$

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Key Concepts

Cartesian Coordinate System

This system represents points as ordered pairs (x, y) in a two?dimensional plane and provides the framework for plotting geometric figures. It allows one to analyze spatial relationships and use algebraic methods to determine positions and distances between points.

Properties of Rectangles

Rectangles are quadrilaterals with opposite sides parallel and congruent, and each of the four interior angles is 90 degrees. Understanding these properties is essential for verifying that a given quadrilateral is indeed a rectangle and for determining missing vertices by ensuring the conditions of parallelism and perpendicularity are met.

Determining Missing Vertices

In coordinate geometry, missing vertices of polygons can be found by using the properties of the shape. For rectangles, since opposite sides are parallel and equal in length, one can deduce the fourth vertex by aligning the known x- and y-coordinates appropriately in agreement with the rectangle's orientation.

Determining if a Rectangle is a Square

Although both squares and rectangles have right angles, a square is a special type of rectangle where all four sides are equal in length. Verifying if a rectangle is also a square involves comparing the lengths of its sides using the distance formula to ensure they are identical.

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