A right-angled triangle has side lengths 12 cm, 16 cm, and 20 cm.\nA new triangle is formed by joining the midpoints of all three sides (the medial triangle). The same midpoint-joining process is repeated on the newly formed triangle, and this continues infinitely.\nFind the sum of the areas (in sq cm) of all triangles formed in this infinite process (including the original triangle).

Introduction:
This question tests two ideas together: (1) area of a right triangle using base and height, and (2) how joining midpoints of a triangle creates a similar triangle with a predictable area ratio. The triangle formed by joining midpoints is called the medial triangle. It is similar to the original triangle, with each side half the original length. Since area scales as the square of the scale factor, the medial triangle’s area becomes one-fourth of the previous triangle’s area. Repeating the process creates an infinite geometric series of areas. The goal is to compute the total sum of that infinite series in a controlled, step-by-step way.

Given Data / Assumptions:

• Right triangle sides: 12 cm, 16 cm, 20 cm
• Right angle is between 12 cm and 16 cm
• Area of right triangle = (1/2) * base * height
• Midpoint-joining creates a similar triangle with scale factor 1/2
• Area scale factor = (1/2)^2 = 1/4 per step

Concept / Approach:
Compute the original area first. Then treat the repeated areas as a geometric series: A + A/4 + A/16 + ... with common ratio r = 1/4. Infinite sum = A / (1 - r). This is valid because |r| < 1 so the series converges.

Step-by-Step Solution:
Original area A0 = (1/2) * 12 * 16 = 96 sq cmEach medial triangle area is 1/4 of the previous: A1 = 96/4 = 24Next: A2 = 24/4 = 6, and so onSo total area sum = 96 + 24 + 6 + ...This is a geometric series with first term a = 96 and common ratio r = 1/4Sum to infinity S = a / (1 - r) = 96 / (1 - 1/4)S = 96 / (3/4) = 96 * (4/3) = 128 sq cm

Verification / Alternative Check:
Partial sums approach 128: 96 + 24 = 120, plus 6 = 126, plus 1.5 = 127.5, plus 0.375 = 127.875, clearly converging to 128. This confirms the infinite series result is sensible and consistent with repeated quartering of area.

Why Other Options Are Wrong:
312 or 412: far too large; would require areas increasing, not shrinking.246 or 192: typically come from adding perimeters or using wrong ratio (like 1/2 instead of 1/4 for area).

Common Pitfalls:
Using scale factor 1/2 for area instead of 1/4.Forgetting to include the original triangle area in the sum.Using the hypotenuse (20 cm) as base/height in right-triangle area.Applying finite-series formula incorrectly to an infinite series.

Final Answer:
128 sq cm