Proof of a Rhombus by Construction and Deduction
Object of Learning
How to prove that a quadrilateral is a rhombus in two ways: by geometric construction and by deductive reasoning.
Definitions
1. A rhombus is a quadrilateral with sides of equal length.
2. Geometric construction is the process of creating geometric shapes using only a compass and a straight edge (ruler) without numerical measurements.
3. Proving by geometric construction involves demonstrating the truth of a geometric statement using a compass and straightedge. These constructions rely on logical deductions and established geometric principles.
4. Proving by deductive reasoning involves proving geometric statements using a logical sequence of steps based on previously accepted facts, definitions, postulates, and theorems
Problem 1
Two congruent circles with centers A and B intersect at points D and C. Prove that ACBD is a rhombus.
Proof by Construction
The construction uses the point tool 
, the line segment tool 
, and the circle through center and point tool
which is the same as the compass tool 
to construct your rhombus ACBD.
, the line segment tool , and the circle through center and point tool which is the same as the compass tool to construct your rhombus ACBD.Practice how to construct a rhombus
What guarantees that AC and AD are the same length?
Why do BC and BD have equal lengths?
How about AC and BC? Why can they have the same lengths?
Proof by deductive reasoning
Answer these problems and enhance your skill in proving by deductive reasoning in statement-reason form.
Problem 2
Given a quadrilateral PQRS. PQ and PR are radii of Circle P, and RQ and RS are radii of Circle R. If Circle P and Circle R are congruent, prove that quadrilateral PQRS is a rhombus.
Problem 3
Refer to Problem 2. If Circle P and Circle R are not congruent, prove that quadrilateral PQRS is a kite.