Chapter 6, Problem 15PFA

a.

To determine

To write the system of equations to find the length of the side of each equilateral triangle.

Answer to Problem 15PFA

  $\begin{array}{l}x+y=16\\ x-y=2\end{array}$

Explanation of Solution

Given information :

The sum of the perimeters of the two different equilateral triangles are 48 inches.

The difference of the perimeters of the two different equilateral triangles are 6 inches.

Concept used :

In equilateral triangles, all three sides are equal.

The perimeter of any triangles is the sum of the length of their three sides.

If a , b and c are the sides of the any triangle then:

The perimeter of the triangle $=a+b+c$

Calculation :

Let:

The length of the side of the first equilateral triangle is x.

And the length of the side of the second equilateral triangle is y.

Perimeter of the first equilateral triangle = $x+x+x=3x$

Perimeter of the second equilateral triangle = $y+y+y=3y$

So, according to the question:

Perimeter of the first equilateral triangle + Perimeter of the second equilateral triangle = 48

  $\begin{array}{l}3x+3y=48\\ \Rightarrow 3(x+y)=48\\ \Rightarrow x+y=\frac{48}{3}\end{array}$

  $\Rightarrow x+y=16$ ……Equation 1

And

Perimeter of the first equilateral triangle - Perimeter of the second equilateral triangle = 6

  $\begin{array}{l}3x-3y=6\\ \Rightarrow 3(x-y)=6\\ \Rightarrow x-y=\frac{6}{3}\end{array}$

  $\Rightarrow x-y=2$ ……. Equation 2

b.

To determine

To find the perimeters of the two triangles.

Answer to Problem 15PFA

Perimeter of the first equilateral triangle = 27 inches.

Perimeter of the second equilateral triangle = 21 inches.

Explanation of Solution

Given information :

The sum of the perimeters of the two different equilateral triangles are 48 inches.

The difference of the perimeters of the two different equilateral triangles are 6 inches.

Formula used :

In equilateral triangles, all three sides are equal.

The perimeter of any triangles is the sum of the length of their three sides.

If a , b and c are the sides of the any triangle then:

The perimeter of the triangle $=a+b+c$

Calculation :

The system of equations is

  $\begin{array}{l}x+y=16\\ x-y=2\end{array}$ .

Adding both the equations gives:

  $\begin{array}{l}(x+y)+(x-y)=16+2\\ \Rightarrow x+y+x-y=18\\ \Rightarrow 2x=18\\ \Rightarrow x=\frac{18}{2}\\ \Rightarrow x=9\end{array}$

Put the value of x in $x+y=16$ and solve for y

  $\begin{array}{l}9+y=16\\ \Rightarrow y=16-9\\ \Rightarrow y=7\end{array}$

The length of the side of the first equilateral triangle is 9inches

And the length of the side of the second equilateral triangle is 7 inches

Perimeter of the first equilateral triangle = $x+x+x=3x$ = $3\times 9=27$ inches

Perimeter of the second equilateral triangle = $y+y+y=3y$ = $3\times 7=21$ inches