What you’ll learn
Prove rhombus properties.
Key idea
Squared distances suffice for equality and avoid rounding square roots.
Equal side lengths, not diagonal equality, define a rhombus.
Worked example
Complete missing statements or reasons in a proof for 'Prove rhombus properties'.
Consecutive coordinate vertices are A=(-2, 7), B=(3, 1), C=(-2, -5), D=(-7, 1). Complete a coordinate proof that ABCD is a rhombus.
- AB^2=BC^2=CD^2=DA^2=5^2+6^2=61.
- All four side lengths are equal, so the convex quadrilateral is a rhombus; its diagonal lengths may differ.
Answer: Four sides are congruent — AB^2=BC^2=CD^2=DA^2=5^2+6^2=61.
Quick Check
Complete missing statements or reasons in a proof for 'Prove rhombus properties'.
Consecutive coordinate vertices are A=(2, 2), B=(4, -1), C=(2, -4), D=(0, -1). Complete a coordinate proof that ABCD is a rhombus.