An isosceles triangle has perimeter 14 cm. Each lateral side is to the base in the ratio 5 : 4. Find the area of the triangle.

Introduction / Context:
For an isosceles triangle with equal lateral sides and base in a given ratio, the exact side lengths can be found by introducing a scale factor. Once side lengths are known, the altitude is obtained via the Pythagorean relation on half the base, and area follows as 1/2 * base * altitude.

Given Data / Assumptions:

• Lateral side : base = 5 : 4
• Perimeter = 14 cm
• Let lateral sides be 5k and 5k, and base be 4k

Concept / Approach:
Use perimeter to find k, then compute the altitude using h = √( (5k)^2 − (4k/2)^2 ). The area is (1/2)*base*h. All arithmetic is exact for integer k here.

Step-by-Step Solution:
Perimeter: 5k + 5k + 4k = 14k = 14 ⇒ k = 1Sides: 5, 5, 4Altitude to base: h = √(5^2 − (4/2)^2) = √(25 − 4) = √21Area = (1/2) * 4 * √21 = 2√21 cm^2

Verification / Alternative check:
Heron’s formula with sides 5, 5, 4 gives area √(7*(7−5)*(7−5)*(7−4)) = √(7*2*2*3) = √84 = 2√21, confirming the same result.

Why Other Options Are Wrong:
1/2√21 and 3/2√21 are too small; “√212” is malformed and does not represent the correct area; only 2√21 cm^2 is consistent with exact computation.

Common Pitfalls:
Using the full base rather than the half-base when applying Pythagoras, or misinterpreting the ratio as 5:4:4 for three different sides. Here two sides are equal (isosceles).

Final Answer:
2√21 cm2