Infinite midpoint-triangle process: Starting with a right triangle with sides 12 cm, 16 cm, and 20 cm, a new right triangle is formed by joining midpoints of the sides; the process continues infinitely. Find the sum of the areas of all triangles formed (including the original).

Introduction / Context:
Joining midpoints of a triangle’s sides forms the medial triangle, whose area is one-quarter of the original. Repeating this generates a geometric series of areas.

Given Data / Assumptions:

• Original right triangle sides: 12 cm, 16 cm, 20 cm
• Area of right triangle = (1/2) * product of perpendicular sides
• Each medial triangle has area = 1/4 of its parent
• We include the original triangle unless otherwise specified (standard for “all triangles so made”).

Concept / Approach:
The sum S of an infinite geometric series with first term a and ratio r (|r| < 1) is S = a / (1 - r). Here a = initial area, r = 1/4.

Step-by-Step Solution:

Verification / Alternative check:
Partial sums: 96 + 24 = 120; + 6 = 126; + 1.5 = 127.5; approaching 128, confirming the limit.

Why Other Options Are Wrong:
312, 412, 246 are unrelated totals; 96 is just the first area without the infinite continuation.

Common Pitfalls:
Confusing side-halving (which halves length) with area-halving; area reduces by a factor of 1/4, not 1/2.

Final Answer:
128 sq.cm