A parallelogram has diagonals that bisect each other. A student concludes it must be a rhombus. What is wrong with this reasoning?
ANothing — bisecting diagonals are the defining property of a rhombus
BBisecting diagonals are a property of all parallelograms, not just rhombuses; perpendicular diagonals are what make a parallelogram a rhombus
CThe student should check for equal angles instead of diagonals
DBisecting diagonals only apply to rectangles, not rhombuses
All parallelograms have diagonals that bisect each other — that property is inherited from the parallelogram definition and distinguishes nothing. What makes a parallelogram a rhombus is that its diagonals are perpendicular. Equivalently, if all four sides are equal, the diagonals must meet at 90°. Bisecting diagonals alone tell you only that you have a parallelogram.
Question 2 Multiple Choice
A rhombus has diagonals of length 6 cm and 8 cm. What is its area?
A48 cm²
B24 cm²
C28 cm²
D14 cm²
Area of a rhombus = (1/2)d₁d₂ = (1/2)(6)(8) = 24 cm². This formula works because the two perpendicular diagonals divide the rhombus into four right triangles, each with legs d₁/2 and d₂/2. Their combined area is 4 × (1/2)(3)(4) = 24 cm². The distractor 48 cm² comes from forgetting the factor of 1/2.
Question 3 True / False
A rhombus can seldom have right angles.
TTrue
FFalse
Answer: False
False. A square is a special case of a rhombus — it has all four sides equal AND all four angles equal to 90°. The rhombus family includes both the 'diamond' shapes with acute and obtuse angles AND the square with right angles. Thinking a rhombus must have non-right angles confuses the typical visual appearance with the definition.
Question 4 True / False
The perpendicularity of a rhombus's diagonals is a direct consequence of all four sides being equal.
TTrue
FFalse
Answer: True
True. The proof uses SSS congruence: label the rhombus ABCD with diagonals meeting at E. Triangles ABE and ADE share side AE, and have AB = AD (equal sides) and BE = DE (diagonals bisect each other from parallelogram properties). SSS congruence means the angles at E are both equal and supplementary, forcing each to be 90°. The equal sides are precisely what creates this constraint.
Question 5 Short Answer
Why does adding the constraint 'all four sides equal' to a parallelogram force its diagonals to be perpendicular?
Think about your answer, then reveal below.
Model answer: Equal sides create congruent triangles on either side of the diagonal (by SSS), which means the angles where the diagonal meets the other diagonal must be both equal and supplementary — forcing them to be 90°.
The argument: in rhombus ABCD, let the diagonals meet at E. Triangles ABE and ADE have three equal sides (AB = AD by definition, AE = AE shared, BE = DE because diagonals bisect each other in any parallelogram). SSS congruence means angle AEB = angle AED. Since these angles are also supplementary (they form a straight line), each must be 90°. This is why perpendicularity is a consequence of equal sides, not an independent assumption.