Chapter 15, Problem 9RP

To determine

To list $\overline{\mathrm{A}\mathrm{B}},\overline{\mathrm{A}\mathrm{C}}$ and $\overline{\mathrm{B}\mathrm{C}}$ in order of size from longest to shortest.

Answer to Problem 9RP

  $\mathrm{B}\mathrm{C},\mathrm{A}\mathrm{C},\mathrm{A}\mathrm{B}$ are the size from longest to shortest.

Explanation of Solution

Given:

The given is the following triangle-

  

Calculation:

Using the Triangle Exterior-Angle theorem,

  $\begin{array}{l}\mathrm{x}+100=(6\mathrm{x}-50)+(2\mathrm{x}+10)\\ \mathrm{x}+100=8\mathrm{x}-40\\ -7\mathrm{x}=-140\\ \mathrm{x}=\frac{140}{7}\\ \mathrm{x}=20\end{array}$

Substitute $\mathrm{x}$ into the equation of $\mathrm{m}\angle \mathrm{A}$ ,

  $\begin{array}{l}\mathrm{m}\angle \mathrm{A}=6(20)-50=70\\ \mathrm{m}\angle \mathrm{C}=2(20)+10=50\\ \mathrm{m}\angle \mathrm{A}+\mathrm{m}\angle \mathrm{B}+\mathrm{m}\angle \mathrm{C}=180\\ 70+\mathrm{m}\angle \mathrm{B}+50=180\\ \mathrm{m}\angle \mathrm{B}=180-120=60\\ \mathrm{m}\angle \mathrm{B}=60\end{array}$

The sides will be according to the angles i.e. for larger angle side will be larger and for smaller, side will be smaller.

  $\mathrm{B}\mathrm{C},\mathrm{A}\mathrm{C},\mathrm{A}\mathrm{B}$ are the size from longest to shortest.