Question

The areas of two similar triangles are 18 cm2 and 8 cm2. One of the sides of the first triangle is 4.5 cm. What is the length of the corresponding side of the other triangle?

Answer 1

The corresponding side is 3 cm

Explanation:

When we have area, it is the related to the scale factor by the scale factor squared

Area large/ area of small = 18/8 = 9/4

Take the square root

sqrt(9/4) = 3/2

The scale factor is 3/2 large to small

The large side is 4.5 cm

large 3 4.5 cm

-------- = ------ = -----------

small 2 x cm

Using cross products

3x = 2(4.5)

3x =9

Divide each side by 3

3x/3 = 9/3

x =3

Answer 2

`\bf ~\hspace{5em} \textit{ratio relations of two similar shapes} \\[2em] \begin{array}{ccccllll} &\stackrel{ratio~of~the}{Sides}&\stackrel{ratio~of~the}{Areas}&\stackrel{ratio~of~the}{Volumes}\\ \cline{2-4}&\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array} \\\\---------------------------------`

`\bf \cfrac{\textit{similar shape}}{\textit{similar shape}}\qquad \cfrac{s}{s}=\cfrac{√(s^2)}{√(s^2)}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{large}{small}\qquad \qquad \cfrac{side^2}{side^2}=\cfrac{area}{area}\implies \cfrac{4.5^2}{s^2}=\cfrac{18}{8}\implies \cfrac{4.5^2}{s^2}=\cfrac{9}{4}`

`\bf \left( \cfrac{4.5}{s} \right)^2=\cfrac{3^2}{2^2}\implies \cfrac{4.5}{s}=\sqrt{\cfrac{3^2}{2^2}}\implies \cfrac{4.5}{s}=\cfrac{3}{2}\implies 9=3s \\\\\\ \cfrac{9}{3}=s\implies 3=s`