Question

a pair of similar figures is given below. (note: the figures are not drawn to scale.)
the sides \\(\overline{bc}\\) and \\(\overline{de}\\) are corresponding sides.
use the information below to find \\(bc\\) in figure 1 and the perimeter of figure 2.

figure 1	figure 2

| \\(\

$$\begin{tikzpicture}scale=0.5\ \\coordinate (b) at (-1,0);\ \\coordinate (c) at (1,-1);\ \\draw (b) -- (0,1) -- (1,0.5) -- (c) -- cycle;\ \ ode at (-1.5,0) {b};\ \ ode at (1.5,-1) {c};\ \ ode at (-0.5,-2) {area of figure 1 = \\(75\\,\\text{m}^2\\)};\ \ ode at (-0.5,-2.5) {perimeter of figure 1 = \\(20\\,\\text{m}\\)};\ \ ode at (-0.5,-3) {\\(bc = \\square\\,\\text{m}\\)};\ \\end{tikzpicture}$$

\\) | \\(\

$$\begin{tikzpicture}scale=0.5\ \\coordinate (d) at (-1,0);\ \\coordinate (e) at (1,-1);\ \\draw (d) -- (0,2) -- (2,1.5) -- (1.5,0) -- (e) -- cycle;\ \ ode at (-1.5,0) {d};\ \ ode at (1.5,-1) {e};\ \ ode at (0.5,-2) {area of figure 2 = \\(243\\,\\text{m}^2\\)};\ \ ode at (0.5,-2.5) {perimeter of figure 2 = \\(\\square\\,\\text{m}\\)};\ \ ode at (0.5,-3) {\\(de = 9\\,\\text{m}\\)};\ \\end{tikzpicture}$$

\\) |

Explanation:

Step1: Find area ratio

Area ratio = $\frac{\text{Area of Figure 1}}{\text{Area of Figure 2}} = \frac{75}{243} = \frac{25}{81}$

Step2: Calculate scale factor

Scale factor $k = \sqrt{\frac{25}{81}} = \frac{5}{9}$ (since $k>0$)

Step3: Compute BC

$BC = DE \times k = 9 \times \frac{5}{9} = 5$ m

Step4: Find perimeter of Figure 2

Perimeter ratio = $k$, so $\text{Perimeter of Figure 2} = \frac{\text{Perimeter of Figure 1}}{k} = 20 \div \frac{5}{9} = 36$ m

Answer:

BC = 5 m
Perimeter of Figure 2 = 36 m