In an isosceles triangle, each of the two equal sides is twice as long as the third side. An equilateral triangle is constructed such that its perimeter is the same as that of the isosceles triangle. What is the ratio of the area of the isosceles triangle to the area of the equilateral triangle?

Introduction:
This geometry problem compares the areas of two different triangles that share the same perimeter: one is isosceles with a specific side relationship, and the other is equilateral. It tests knowledge of triangle area formulas, height computation, and careful ratio simplification.

Given Data / Assumptions:

• Isosceles triangle: equal sides each have length 2a, base has length a.
• Perimeter of isosceles triangle = 2 * 2a + a = 5a.
• Equilateral triangle has the same perimeter, so each side is 5a / 3.
• We must find area(isosceles) : area(equilateral).

Concept / Approach:
For the isosceles triangle, we find its height using the Pythagorean theorem by dropping a perpendicular from the apex to the base, splitting the base into two segments of length a/2. For the equilateral triangle, we use the standard area formula in terms of side. After computing both areas in terms of a, we form a ratio and simplify.

Step-by-Step Solution:
Isosceles triangle sides: 2a, 2a, a. Drop altitude to base a, splitting it into a/2 and a/2. Height hᵢ = √[(2a)² − (a/2)²] = √[4a² − a²/4] = √[(16a² − a²)/4] = √(15a²/4) = (a/2)√15. Area isosceles Aᵢ = (1/2) * base * height = (1/2) * a * (a/2)√15 = (a²/4)√15. Equilateral triangle side s = 5a / 3. Area equilateral Aₑ = (√3 / 4) * s² = (√3 / 4) * (25a² / 9) = (25√3 a²) / 36. Form ratio Aᵢ : Aₑ = [(a²√15)/4] : [(25√3 a²)/36]. Cancel a² and simplify: = (√15 / 4) * (36 / 25√3) = (36 / 100) * (√15 / √3). Compute √15 / √3 = √(15/3) = √5. So ratio = (36 / 100)√5 = 36√5 : 100.

Verification / Alternative check:
You can choose a convenient value, for example a = 1, compute both areas numerically with that choice, and confirm that their ratio simplifies to 36√5 : 100. This matches the algebraic derivation and confirms the correctness of the simplification.

Why Other Options Are Wrong:
Ratios 30√5 : 100, 32√5 : 100, or 42√5 : 100 correspond to using incorrect heights or incorrect equilateral area formulas. Only 36√5 : 100 arises from the correct application of the Pythagorean theorem and standard area formulas.

Common Pitfalls:
Errors usually occur in computing the isosceles triangle height or in misusing the equilateral triangle area formula, such as omitting the √3 factor or mishandling fractions when simplifying. Being systematic with algebraic steps prevents these mistakes.

Final Answer:
The ratio of the areas is 36√5 : 100.