Chapter 5.9, Problem 15IP

(a)

To determine

To find: The expression for the perimeter of $\Delta ABC$ .

Answer to Problem 15IP

The expression for the perimeter of $\Delta ABC$ is $\text{Perimeter}=a+b+c$ .

Explanation of Solution

Given information:

The given triangles are shown below and $\Delta XYZ\sim \Delta ABC$ .

  

Calculation:

Perimeter is defined as the outside distance of a two-dimensional shape. Let the perimeter of $\Delta ABC$ is $P_1$ .

Write the expression for the perimeter of $\Delta ABC$ .

  $\begin{array}{c}{P}_1=AB+BC+CA\\ =c+a+b\\ =a+b+c\end{array}$

Therefore, the expression for the perimeter of $\Delta ABC$ is $\text{Perimeter}=a+b+c$ .

(b)

To determine

To find: The expression for the measure of the sides of $\Delta XYZ$ .

Answer to Problem 15IP

The expression for the measure of the sides of $\Delta XYZ$ are $XY=d\cdot c$ , $YZ=d\cdot a$ , and $ZX=d\cdot b$ .

Explanation of Solution

Given information:

The given triangles are shown below and $\Delta XYZ\sim \Delta ABC$ . The scale factor is $d$ .

  

Calculation:

Scale factor is defined as the compare of two lengths or sizes. Let $\Delta XYZ$ is larger than $\Delta ABC$ .

Since, $\Delta XYZ$ is larger than $\Delta ABC$ and $\Delta XYZ\sim \Delta ABC$ . So, the sides of $\Delta XYZ$ will be multiple of the scale factor $d$ .

Write the expression for the sides of $\Delta XYZ$ .

  $\begin{array}{c}XY=d\cdot AB\\ XY=d\cdot c\end{array}$

and,

  $\begin{array}{c}YZ=d\cdot BC\\ YZ=d\cdot a\end{array}$

and,

  $\begin{array}{c}ZX=d\cdot CA\\ ZX=d\cdot b\end{array}$

Therefore, the expression for the measure of the sides of $\Delta XYZ$ are $XY=d\cdot c$ , $YZ=d\cdot a$ , and $ZX=d\cdot b$ .

(c)

To determine

To find: The expression for the perimeter of $\Delta XYZ$ .

Answer to Problem 15IP

The expression for the perimeter of $\Delta XYZ$ is $\text{Perimeter}=d\left(a+b+c\right)$ .

Explanation of Solution

Given information:

The given triangles are shown below and $\Delta XYZ\sim \Delta ABC$ . The scale factor is $d$ .

  

Calculation:

From part (b), the measure of the sides of $\Delta XYZ$ are $XY=d\cdot c$ , $YZ=d\cdot a$ , and $ZX=d\cdot b$ . Let the perimeter of $\Delta XYZ$ is $P_2$ .

Write the expression for the perimeter of $\Delta XYZ$ .

  $\begin{array}{c}{P}_2=XY+YZ+ZX\\ =d\cdot c+d\cdot a+d\cdot b\\ =d\left(a+b+c\right)\end{array}$

Therefore, the expression for the perimeter of $\Delta XYZ$ is $\text{Perimeter}=d\left(a+b+c\right)$ .

(d)

To determine

To explain: The meaning of the expression for the perimeter of $\Delta XYZ$ using distributive property of factor.

Answer to Problem 15IP

The product of the scale factor with the sum of the sides is equal to the sum of the product of the scale factor with individual sides.

Explanation of Solution

Given information:

The given triangles are shown below and $\Delta XYZ\sim \Delta ABC$ . The scale factor is $d$ .

  

Calculation:

From part (c), the expression for the perimeter of $\Delta XYZ$ is $\text{Perimeter}=d\left(a+b+c\right)$ .

Consider the distributive property of factor.

  $d\left(a+b+c\right)=d\cdot a+d\cdot b+d\cdot c$

From the above, it is clear that the product of the scale factor with the sum of the sides is equal to the sum of the product of the scale factor with individual sides.

(e)

To determine

To find: The perimeter of $\Delta XYZ$ .

Answer to Problem 15IP

The perimeter of $\Delta XYZ$ is 24 inches.

Explanation of Solution

Given information:

The triangles $\Delta XYZ\sim \Delta ABC$ . The scale factor is 2. The measure of the sides are $AB=3\text{inches}$ , $BC=4\text{inches}$ and $AC=5\text{inches}$ .s

  

Calculation:

From part (c), the expression for the perimeter of $\Delta XYZ$ is,

  $P_2=d\left(AB+BC+CA\right)$

Substitute the values in the above expression.

  $\begin{array}{c}{P}_2=2\left(3+4+5\right)\\ =2\times 12\\ =24\text{inches}\end{array}$

Therefore, the perimeter of $\Delta XYZ$ is 24 inches.

(f)

To determine

To explain: The significance of the scale factor with the perimeter of similar figures.

Answer to Problem 15IP

The significance of the scale factor with the perimeter of similar figures is given below.

Explanation of Solution

Calculation:

Scale factor is defined as the ratio of two lengths or sizes. The perimeter of the similar figures depends upon the scale factor. If the scale factor is greater than 1, the perimeter of the larger figure is more and vice versa.