Question

The hypotenuese of an isosceles right triangle is 16 inches. The midpoints of its sides are connected to form an inscribed triangle, and this process is repeated, creating a third triangle inscribed in the previous one. Find the sum of the areas of these triangles if this process is continued infinitely.

Answer 1

256/3 = 85 1/3 square inches

Explanation:

The dimensions of the first inscribed triangle are 1/2 those of the original, so its area is (1/2)² = 1/4 of the original. The area of the original is ...

A = (1/2)bh = (1/2)(16/√2)(16/√2) = 64 . . . . square inches

The sum of an infinite series with first term 64 and common ratio 1/4 is ...

S = a1/(1 -r) . . . . . . for first term a1 and common ratio r

= 64/(1 -1/4) = 64(4/3) = 256/3 . . . . square inches

The sum of the areas of the triangles is 256/3 = 85 1/3 square inches.