Question

Triangles $ABC$ and $DEF$ are similar.

(i) If area ($△$ ABC)= $16$ $c{m}^2$, area($△$ DEF)= $25$ $c{m}^2$ and BC = $2.34$ cm, find EF.

(ii) If area ($△$ ABC)= $9$ $c{m}^2$, area($△$ DEF) = $64$ $c{m}^2$ and DE= $5.1$ cm, find AB.

(iii) If AC = $19$ cm and DF=$8$ cm, ratio of the area of two triangles.

(iv) If area ($△$ ABC)= $36$ $c{m}^2$, area ($△$ DEF) =$64$ $c{m}^2$ and DE= $6.2$ cm, find AB.

(v) If AB= $1.2$ cm and DE=$1.4$ find the ratio of the areas of triangle ABC and DEF.

Answer 1

$\left(i\right)$ We know that the ratio of area of two similar triangles is equal to the ratio of squares of their corresponding sides.

$\therefore$ $\frac{area\left(△ABC\right)}{area\left(△DEF\right)}={\left(\frac{BC}{EF}\right)}^2$

$\Rightarrow$ $\frac{16}{25}={\left(\frac{2.3}{EF}\right)}^2$

$\Rightarrow$ $\frac{4}{5}=\frac{2.3}{EF}$

$\Rightarrow$ $EF=\frac{2.3\times 5}{4}$

$\therefore$ $EF=2.875cm$

$\left(ii\right)$ We know that the ratio of area of two similar triangles is equal to the ratio of squares of their corresponding sides.

$\therefore$ $\frac{area\left(△ABC\right)}{area\left(△DEF\right)}={\left(\frac{AB}{DE}\right)}^2$

$\Rightarrow$ $\frac{9}{64}={\left(\frac{AB}{DE}\right)}^2$

$\Rightarrow$ $\frac{3}{8}=\frac{AB}{5.1}$

$\Rightarrow$ $AB=\frac{5.1\times 3}{8}$

$\therefore$ $AB=1.91cm$

$\left(iii\right)$ We know that the ratio of area of two similar triangles is equal to the ratio of squares of their corresponding sides.

$\therefore$ $\frac{area\left(△ABC\right)}{area\left(△DEF\right)}={\left(\frac{AC}{DF}\right)}^2$

$\therefore$ $\frac{area\left(△ABC\right)}{area\left(△DEF\right)}={\left(\frac{19}{8}\right)}^2$

$\therefore$ $\frac{area\left(△ABC\right)}{area\left(△DEF\right)}=\left(\frac{361}{64}\right)$

$\left(iv\right)$ We know that the ratio of area of two similar triangles is equal to the ratio of squares of their corresponding sides

$\therefore$ $\frac{area\left(△ABC\right)}{area\left(△DEF\right)}={\left(\frac{AB}{DE}\right)}^2$

$\Rightarrow$ $\frac{36}{64}={\left(\frac{AB}{DE}\right)}^2$

$\Rightarrow$ $\frac{6}{8}=\frac{AB}{6.2}$

$\Rightarrow$ $AB=\frac{6.2\times 6}{8}$

$\therefore$ $AB=4.65cm$

$\left(v\right)$ We know that the ratio of area of two similar triangles is equal to the ratio of squares of their corresponding sides.

$\therefore$ $\frac{area\left(△ABC\right)}{area\left(△DEF\right)}={\left(\frac{AB}{DE}\right)}^2$

$\therefore$ $\frac{area\left(△ABC\right)}{area\left(△DEF\right)}={\left(\frac{1.2}{1.4}\right)}^2$

$\therefore$ $\frac{area\left(△ABC\right)}{area\left(△DEF\right)}=\frac{36}{49}$