Chapter 11.6, Problem 4P

To determine

To calculate:

Sum of perimeter of the triangles.

Answer to Problem 4P

Sum of perimeter of the triangles is $120$ .

Explanation of Solution

Give information:

Side of equilateral triangle is $10cm$

Calculation:

Mid-points of its sides are joined to from another triangle whose mid points are joined to from another triangle.

This process continues indefinitely.

Side of second equilateral triangle $=10cm$

(line segment joining mid-point of two sides of a triangle is parallel to third side and half of it)

Side of third equilateral triangle $=5cm$

(sides are $3,1,5,\frac{3}{4}$ )

Perimeter of first triangle $=20+20+20=60cm$

Perimeter of second triangle $=10+10+10=30cm$

Perimeter of third triangle $=5+5+5=15cm$

Sum of perimeter of all triangles $=60+30+15+9+\frac{9}{2}+\frac{9}{4}$

This is geometric progression series where,

Common ratio,

  $=\frac{30}{60}=\frac{1}{2}$

Sum of an infinity geometric progression,

  $=\frac{a}{1-r}$

Where,

  $a=$ initial term

  $r=$ common ratio

  $\begin{array}{l}=\frac{60}{1-\frac{1}{2}}\\ =60\times 2\\ =120\end{array}$

Therefore, Sum of perimeter of the triangles is $120$ .