Question

Geometry question

Answer 1

An equilateral triangle is drawn . then an incircle is drawn, then an equilateral triangle is drawn , then an incircle is drawn upto infinite times , The side of the outer equilateral triangle is \(10\) .
Find the area of the equilateral trianagles upto infinity so formed

Answer 2

The first step is to find the ratio of the initial area to the area of the second triangle.

Answer 3

correction: find the sum of area 's of triangles upto infinty so formed ?

Answer 4

Summing up all of the areas? That's still the first step.

To do that, start by finding the height and area of the initial triangle. But, instead of using 10, use a variable L to keep things clean for now.

Answer 5

\(A_1=\dfrac{\sqrt{3}l^2}{4}\)

Answer 6

OK.
Do you know the height of the triangle? That will be helpful in a minute.

Answer 7

\(h=\dfrac{\sqrt3l}{2}\)

Answer 8

Equilateral triangles have this nifty property: The distance from the center of a side to the center of the triangle is 1/3 the height of the triangle.