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Chapter 1: Problem 57

Determine whether the statement is true or false. Justify your answer. If four points represent the vertices of a polygon, and the four sides are equal, then the polygon must be a square.

Short Answer

Expert verified
The statement is false. A four-sided polygon with all sides equal could also be a non-square rhombus, not only a square.

Step by step solution

01

Understanding quadrilaterals with equal sides

If a quadrilateral has four sides of equal length, it qualifies as a rhombus. A rhombus is a quadrilateral whose four sides all have the same length. However, a rhombus doesn't necessarily have to have all angles equal to 90 degrees.
02

Contradicting the statement

The given statement says that if a quadrilateral has four equal sides, then it must be a square. This is a universal generalization and to disprove it, a counter-example should be enough. A rhombus with a non-90-degree angle serves as a counter example in this context. For such a rhombus, all four sides are equal but it is not a square.
03

Concluding the falseness of the statement

Since we've found a counter-example where all four sides of a quadrilateral are equal but the shape is not a square, we can conclude that the given statement is false. Therefore, it is not true that if a polygon has four points which represent the vertices and four sides are equal, then the polygon must be a square.

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