Chapter 14.4, Problem 6MRE

To determine

To find: Ratio of areas of $\Delta ODE$ to $\Delta OFG$ .

Answer to Problem 6MRE

Ratio of areas of $\Delta ODE$ to $\Delta OFG$ is $9:25$ .

Explanation of Solution

Given information :

  $\overline{ED}\left|\right|\overline{BC}\left|\right|\overline{FG}$

Concept used:

If two objects have the same shape, they are called "similar." When two figures are similar, the ratios of the lengths of their corresponding sides are equal. To determine if the triangles shown are similar, compare their corresponding sides.

As per the given problem

Following diagram is provided.

  

Given: $\overline{ED}\left|\right|\overline{BC}\left|\right|\overline{FG}$

  $\Delta ODE$ similar to $\Delta OFG$ .

  $\begin{array}{l}\frac{OD}{OF}=\frac{DE}{FG}=\frac{EO}{GO}\\ \Rightarrow \frac{OD}{OF}=\frac{x}{3+w}=\frac{\frac{9}{2}}{3+\frac{9}{2}}=\frac{9}{6+9}=\frac{9}{15}=\frac{3}{5}\end{array}$

  $\frac{ar(\Delta OBC)}{ar(\Delta ODE)}=\frac{O{C}^2}{O{F}^2}={(\frac{OD}{OF})}^2$

  $\Rightarrow {(\frac{3}{5})}^2=\frac{9}{25}$

Hence,

Ratio of areas of $\Delta ODE$ to $\Delta OFG$ is $9:25$ .