Chapter 6.5, Problem 15E

a.

To determine

To find: Length of $DE$ and length of $FG$ .

Answer to Problem 15E

Length of $DE=10\text{ cm}$ and length of $FG=12.5\text{ cm}$ .

Explanation of Solution

Given information:

The following figure:

  

  $△ABC,△ADE,△AFG$ are similar.

To find $DE$ and $FG$ , write and solve a proportion.

For $DE$ :

  $\begin{array}{l}\frac{DE}{BC}\text{=}\frac{AD}{AB}\text{ [Write Proportion]}\\ \frac{DE}{5}\text{=}\frac{12+12}{12}\text{ [Substitute values]}\\ \frac{DE}{5}\text{=}\frac{24}{12}\text{ [Add]}\\ \frac{DE}{5}\text{=}\frac{2}{1}\text{ [Divide out common factors]}\\ DE\cdot 1=5\cdot 2\text{ [Form cross product]}\\ DE=10\text{ [Multiply] }\end{array}$

For $FG$ :

  $\begin{array}{l}\frac{FG}{BC}\text{=}\frac{AF}{AB}\text{ [Write Proportion]}\\ \frac{FG}{5}\text{=}\frac{6+12+12}{12}\text{ [Substitute values]}\\ \frac{FG}{5}\text{=}\frac{30}{12}\text{ [Add]}\\ \frac{FG}{5}\text{=}\frac{5}{2}\text{ [Divide out common factors]}\\ FG\cdot 2=5\cdot 5\text{ [Form cross product]}\\ FG\cdot 2=25\text{ [Multiply]}\\ FG=12.5\text{ [Divide both sides by 2]}\end{array}$

Therefore,

  $DE=10\text{ cm}$ and $FG=12.5\text{ cm}$ .

b.

To determine

To find: Length of $AE$ and length of $AG$ .

Answer to Problem 15E

Length of $AE=26\text{ cm}$ and length of $AG=32.5\text{ cm}$ .

Explanation of Solution

Given information:

The following figure:

  

  $△ABC,△ADE,△AFG$ are similar.

To find $AE$ and $AG$ , write and solve a proportion.

For $AE$ :

  $\begin{array}{l}\frac{AE}{AC}\text{=}\frac{AD}{AB}\text{ [Write Proportion]}\\ \frac{AE}{13}\text{=}\frac{12+12}{12}\text{ [Substitute values]}\\ \frac{AE}{13}\text{=}\frac{24}{12}\text{ [Add]}\\ \frac{AE}{13}\text{=}\frac{2}{1}\text{ [Divide out common factors]}\\ AE\cdot 1=13\cdot 2\text{ [Form cross product]}\\ AE=26\text{ [Multiply] }\end{array}$

For $AG$ :

  $\begin{array}{l}\frac{AG}{AC}\text{=}\frac{AF}{AB}\text{ [Write Proportion]}\\ \frac{AG}{13}\text{=}\frac{6+12+12}{12}\text{ [Substitute values]}\\ \frac{AG}{13}\text{=}\frac{30}{12}\text{ [Add]}\\ \frac{AG}{13}\text{=}\frac{5}{2}\text{ [Divide out common factors]}\\ AG\cdot 2=13\cdot 5\text{ [Form cross product]}\\ AG\cdot 2=65\text{ [Multiply]}\\ AG=32.5\text{ [Divide both sides by 2]}\end{array}$

Therefore,

  $AE=26\text{ cm}$ and $AG=32.5\text{ cm}$ .