Chapter 12, Problem 56SGA

a.

To determine

To find: the perimeter of new triangle by using the given information.

Answer to Problem 56SGA

3 units

Explanation of Solution

Given information:

The new triangle formed by joining the midpoints of the sides of an equilateral triangle is also an equilateral triangle.

Perimeter of original triangle = 6 units

  

Calculation:

Evaluate the side $\left(a\right)$ of original triangle as follows:

  $\begin{array}{c}\text{Perimeter}=6\text{ units}\\ 3a=6\text{ units}\\ a=\frac{6\text{ units}}{3}\\ =2\text{ units}\end{array}$

Each side of new triangle is half of the original triangle i.e. $\left(a\text{'}=\frac{a}{2}\right)$ . Now evaluate the perimeter of new triangle as follows:

  $\begin{array}{c}\text{Perimeter}=3a\text{'}\\ =\frac{3a}{2}\\ =3\left(\frac{2}{2}\right)\text{units}\\ =3\text{ units}\end{array}$

Thus the perimeter of new triangle is 3 units.

b.

To determine

To find: the sum of perimeters of all the triangles if the process is continued to form a nested triangles.

Answer to Problem 56SGA

12 units

Explanation of Solution

Given information:

The new triangle formed by joining the midpoints of the sides of an equilateral triangle is also an equilateral triangle.

Perimeter of original triangle = 6 units

Perimeter of new triangle = 3 units

  

Calculation:

Let the perimeter of first triangle be $a_1$ , perimeter of second triangle be $a_2$ and so on. Now evaluate the common ratio $\left(r\right)$ for the geometric sequence.

  $\begin{array}{c}r=a_2÷a_1\\ =3÷6\\ =0.5\end{array}$

The sum of the perimeters of all the triangles is the sum of all terms of the geometric series, i.e. $S_n$ .

  $\begin{array}{c}{S}_n=\frac{a_1}{1-r}\\ =\frac{6}{1-0.5}\left[\text{Put }{a}_1\text{ and }r\right]\\ =\frac{6}{0.5}\\ =12\end{array}$

Thus if the process is continued to form a nested triangles then the sum of perimeters of all the triangles will be 12 units.