Rhombus diagonals – one equals the side: If in a rhombus one of the diagonals is exactly equal to the side length of the rhombus, then what is the ratio of the two diagonals?

Introduction / Context:
A rhombus has all sides equal and diagonals that intersect at right angles. For a rhombus with diagonals p and q and side s, the relationship between side and diagonals comes from the right triangle formed by half-diagonals.

Given Data / Assumptions:

• Side length = s.
• Diagonals are p and q; they bisect each other at right angles.
• One diagonal is equal to the side: without loss of generality, set p = s.

Concept / Approach:
The half-diagonals form a right triangle with hypotenuse s: (p/2)^2 + (q/2)^2 = s^2. Substitute p = s and solve for q in terms of s to obtain the ratio between p and q.

Step-by-Step Solution:
(s/2)^2 + (q/2)^2 = s^2s^2/4 + q^2/4 = s^2 ⇒ q^2/4 = s^2 − s^2/4 = (3/4)s^2q^2 = 3s^2 ⇒ q = s√3Therefore, the diagonals are s and s√3, so the ratio (longer):(shorter) = √3:1.

Verification / Alternative check:
If p = s and q = s√3, plug back into (p/2)^2 + (q/2)^2: s^2/4 + (3s^2)/4 = s^2, which holds.

Why Other Options Are Wrong:
Ratios like √2:1 or 2:1 do not satisfy the Pythagorean relation with p set equal to s. 3:1 overly enlarges q.

Common Pitfalls:
Confusing “one diagonal equals side” with “diagonals equal each other” (which would indicate a square). Here only one diagonal equals s.

Final Answer:
√3 : 1