Isosceles triangle with altitude 8 cm and perimeter 32 cm:\nFind the area of the triangle.

Introduction / Context:
This is an isosceles triangle with a given altitude to the base and a fixed perimeter. Using right-triangle relations after altitude splitting helps determine base and then area.

Given Data / Assumptions:

• Altitude to base = 8 cm.
• Perimeter = 32 cm.
• Let base = b and equal sides = a.

Concept / Approach:
The altitude to the base bisects the base: half-base = b/2. In the right triangle, a^2 = (b/2)^2 + 8^2. Perimeter: b + 2a = 32. Solve for b, then area = (1/2)*b*8.

Step-by-Step Solution:

Verification / Alternative check:
With b = 12, halves are 6; a = √(6^2 + 8^2) = √100 = 10; perimeter = 12 + 20 = 32 (checks).

Why Other Options Are Wrong:
50, 60, 70, 80 do not satisfy both the perimeter and altitude constraints; 48 is exact.

Common Pitfalls:
Assuming altitude equals median to equal sides; here it splits the base and creates two 6–8–10 right triangles.

Final Answer:
48