Starting with an equilateral triangle of side length 3 units, construct a second triangle by connecting the mid points of the sides of the first triangle. Construct a third triangle by connecting the mid points of the sides of the second triangle. If we continue the process indefinitely then sum of the areas of all the triangle is
Answers
Given : Starting with an equilateral triangle of side length 3 units, construct a second triangle by connecting the mid points of the sides of the first triangle. Construct a third triangle by connecting the mid points of the sides of the second triangle.
To Find : sum of the areas of all the triangle if process is continued indefinitely
Solution
Area of Equilateral Triangle = (√3 / 4) side²
Side = 3 cm
Area of 1st Triangle = (√3 / 4) * 3² sq units
A line joining mid point of two sides of triangle is half of third side Hence
2nd triangle side will be 3/2 and third will be 3/4 and so on
Area of 2nd Triangle = (√3 / 4) *(3/2)² sq units
Area of 3rd Triangle = (√3 / 4) *(3/4)² sq units
and so on
Total Area
= (√3 / 4) * 3² + (√3 / 4) *(3/2)² + (√3 / 4) *(3/4)² +.......
= (√3 / 4) * 3² ( 1 + (1/2)² + (1/4)² + ,................... .....)
= (9√3 / 4) ( 1 + 1/4 + 1/16 + ,................... .....)
Infinite GP
a = 1
r = 1/4
S = 1/(1 - 1/4) = 4/3
(9√3 / 4) ( 4/3)
= 3√3 sq units
sum of the areas of all the triangle is 3√3 sq units
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