Consider a triangle of base 1 cm and height 1 cm. On the same base and on the same side, second triangle is drawn with half the previous height. If this process continues as many times as it is possible, what will be the total area of all the triangles thus formed?
Answers
Total area of all the triangle formed = 1 cm²
Given:
Height of the triangle = 1 cm
Base of the triangle = 1 cm
Second triangle is drawn with half the previous height
This process continues as many times as it is possible
To find:
What will be the total area of all the triangles thus formed
Solution:
Given
Height of the triangle = 1 cm
Base of the triangle = 1 cm
The area of triangle = (1/2)×height × base
= (1/2)×1 × 1 = (1/2) cm²
Given second triangle is drawn with half the previous height
The height of 2nd triangle = (1/2)
Area of 2nd triangle = (1/2) (1/2)(1) = (1/2)² cm²
If the process is continued
Height of 3rd triangle = (1/4)
Area of 3rd triangle = (1/2) (1/4)(1) = (1/8) = (1/2)³ cm²
Area of nth triangle = (1/2)ⁿ cm² ... and so on
And the triangles formed will be infinite
If we observe the above areas
(1/2), (1/2)²,(1/2)³, (1/2)⁴ cm² .... (1/2)ⁿ are in a G.P
where first term a = (1/2) and common ratio = 1/2
Sum of infinite terms in G.P = a /(1 - r)
= (1/2) / (1-1/2)
= (1/2) / (1/2)
= 1
Therefore,
Area of the all triangle formed = 1 cm²
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