Chapter 16.5, Problem 2PS

To determine

To find: the ratio of $\frac{FO}{FI}$ and $\frac{HI}{IK}$ from a given triangle $FGHIJK$ .

Answer to Problem 2PS

  $\frac{FO}{FI}=\frac{17}{29}$ and $\frac{HI}{IK}=\frac{8}{9}$ .

Explanation of Solution

Given:

  

  $\frac{HG}{HF}=\frac{4}{9}\text{ and}\frac{FM}{MK}=\frac{2}{3}$ .

Concept used:

If two points balanced the product of the mass and distance from the line of the balance of on point will be equal to the product if the mass and distance from the same line balance of the other point.

  $\begin{array}{l}\frac{w_1}{w_2}=\frac{l_2}{l_1}.\\ w_1\times l_1=w_2\times l_2.\\ w_b=w_1+w_2.\end{array}$

Where $w$ is weight at which the weight is divided or incase of triangle the line which bisect the two weight.

Calculation:

According to the given:

  $\frac{HG}{HF}=\frac{4}{9}\text{ and}\frac{FM}{MK}=\frac{2}{3}$ .

From the ratio from given the line $HG$ provides a weight on $F$ or $w_F=4.$

Similarly, the ratio from the given line $FK$ provides a weight on $F$ or $w_F=3.$

Therefore, the weight on $F$

  $w_F=3\text{ or 4}$ so, take the multiple of both which is $w_F=12$ .

  $\begin{array}{l}{w}_H=9.\\ W_K=8.\end{array}$

Which implies weight on $I$ :

  $w_I=17.$

According to the given diagram:

  $\begin{array}{l}FI=FO+OI.\\ FI=17+12=29.\\ \frac{FO}{FI}=\frac{17}{29}.\end{array}$

Similarly, that of ratio of

  $\frac{HI}{IK}=\frac{8}{9}$ .

Hence, $\frac{FO}{FI}=\frac{17}{29}$ and $\frac{HI}{IK}=\frac{8}{9}$ .