Hello, warriors837!

Triangle is a 30-60-90 triangle, where

Altitude BD is drawn to AC, forming

Altitude DE is drawn to AB, forming

Altitude EF is drawn to AC, forming

Altitude FG is drawn to AB, forming

Altitude GH is drawn to AC, forming

Altitude HI is drawn to AB, forming

Find the ratio of the area of to the area of

I found that the resulting ratio

. . Close, but incorrect.

Note thattriangles are 30-60-90 right triangles.all

In a 30-60-90 triangle, the side opposite 30° is half the hypotenuse.

The side opposite 60° is times the shortest side.

Let , then

The area of is: . .[1]

In

In

In

. . The area of

We have: .

Drawing two altitudes, we have two similar right triangles.

. . The area of the smaller triangle is of the larger.

This procedure is performed three times.

Hence, the area of the final triangle is: . the area of

Therefore: .