Chapter 9.7, Problem 17PSB

a

To determine

To calculate: The measure of unknown given segment i.e. VS.

Answer to Problem 17PSB

The measure of unknown given segment is = $3\sqrt{3}$ units

Explanation of Solution

Given information :

From the figure given

Triangle VST: $\angle \text{VTS}=30°$

Formula used :

In triangle $30°-60°-90°$ sides are in ratio $1:\sqrt{3}:2$

Calculation :

In triangle VST, we have

  $\begin{array}{l}2\text{x}=6\\ \Rightarrow \text{x}=3\\ \Rightarrow \text{SR}=3\text{ units}\\ \therefore \text{VS=3}\sqrt{3}\text{ units}\end{array}$

b

To determine

To calculate: The measure of unknown given segment i.e. ST.

Answer to Problem 17PSB

The measure of unknown given segment is = 9 units

Explanation of Solution

Given information :

From the figure given

Triangle VRT : VR = 6 units; $\angle \text{VRS}=60°$ and $\angle \text{RTV}=30°$

Formula used :

In triangle $30°-60°-90°$ sides are in ratio $1:\sqrt{3}:2$

Calculation :

In triangle VST, we have

  $\begin{array}{l}\text{VT}\\ =2\times \text{VS}\\ =2\times 3\sqrt{3}\mathrm{......}\text{VS}=3\sqrt{3}\mathrm{...}\text{from part a, above}\\ =6\sqrt{3}\end{array}$

c

To determine

To calculate: The measure of unknown given segment i.e. VT.

Answer to Problem 17PSB

The measure of unknown given segment is = $6\sqrt{3}$ units

Explanation of Solution

Given information :

From the figure given

Triangle VRT: VR = 6 units; $\angle \text{VRS}=60°$ and $\angle \text{RTV}=30°$

Formula used :

In triangle $30°-60°-90°$ sides are in ratio $1:\sqrt{3}:2$

Calculation :

In triangle VRS, we have

  $\begin{array}{l}\text{ST}\\ =\text{VS}\times \sqrt{3}\\ =3\sqrt{3}\times \sqrt{3}\mathrm{......}\text{VS}=3\sqrt{3}\mathrm{...}\text{from part a, above}\\ =9\end{array}$

d

To determine

To find: The ratio of perimeters of triangles VSR and VRT

Answer to Problem 17PSB

The ratio of perimeters of given triangles is

Explanation of Solution

Given information :

From the figure given

Triangle VRT: VR = 6 units; $\angle \text{VRS}=60°$ and $\angle \text{RTV}=30°$

Formula used :

In triangle $30°-60°-90°$ sides are in ratio $1:\sqrt{3}:2$

Calculation :

The required ratio

  $\begin{array}{l}=\frac{\text{P(}\Delta \text{VSR)}}{\text{P(}\Delta \text{VRT)}}\\ =\frac{3+6+3\sqrt{3}}{3\sqrt{3}+9+6\sqrt{3}}\\ =\frac{9+3\sqrt{3}}{9+9\sqrt{3}}\\ =\frac{3+\sqrt{3}}{3+3\sqrt{3}}\end{array}$

  $\begin{array}{l}=\frac{1}{3}\left(\frac{3+\sqrt{3}}{1+\sqrt{3}}\right)\\ =\frac{1}{3}\left(\frac{3+\sqrt{3}}{1+\sqrt{3}}\times \frac{1-\sqrt{3}}{1-\sqrt{3}}\right)\\ =\frac{1}{3}\left(\frac{3+\sqrt{3}-3\sqrt{3}-3}{1-3}\right)\\ =\frac{1}{3}\left(\frac{-2\sqrt{3}}{-2}\right)\end{array}$

  $\begin{array}{l}=\frac{\sqrt{3}}{3}=\frac{1}{\sqrt{3}}\\ \Rightarrow \text{P(}\Delta \text{VSR):P(}\Delta \text{VRT)}=1:\sqrt{3}\end{array}$