Similarity : Exercise (8.1) - Solutions

1.           State why the two polygons are, are not, similar.

(a)
The two rectangles are not similar because corresponding sides are not proportional.

(b)
The two squares are similar because corresponding sides are proportional and corresponding angles are equal.

(c)
The two triangles are similar because corresponding sides are proportional and corresponding angles are equal.

(d)
The two polygons are not similar because they have different number of sides.

2.           Complete the proportions.

             (a)    If $\triangle A B C \sim \triangle D E F$ then $\displaystyle\frac{A B}{?}=\frac{B C}{?}=\frac{?}{D F}$.

             (b)     If $\triangle G H I \sim \triangle K L M$ then $\displaystyle\frac{?}{H I}=\frac{?}{G H}=\frac{?}{G I}$.

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(a) $\displaystyle\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$,

(b) $\displaystyle\frac{LM}{HI}=\frac{KL}{GH}=\frac{KM}{GI}$

3.           State whether the proportions are correct for the indicated similar triangles.

           $\begin{array}{l} \text{(a)}\ \triangle A B C \sim \triangle X Y Z.\\\\ \quad\ \ \displaystyle\frac{A B}{X Y}=\displaystyle\frac{B C}{Y Z}\\\\ \text{(b)}\ \triangle D E F \sim \triangle H I J.\\\\ \quad\ \ \displaystyle\frac{D E}{H I}=\displaystyle\frac{E F}{IJ}\\\\ \text{(c)}\ \triangle R S T \sim \triangle L M K.\\\\ \quad\ \ \displaystyle\frac{R T}{L M}=\displaystyle\frac{S T}{M K}\\\\ \text{(d)}\ \triangle X Y Z \sim \triangle U VW.\\\\ \quad\ \ \displaystyle\frac{X Y}{U V}=\displaystyle\frac{X Z}{V W} \end{array}$

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(a) correct

(b) correct

(c) incorrect

(d) correct

4.           Given    :     $\triangle P Q R \sim \triangle U V W$ and lengths of sides are as marked.

Find      :     The values of $x$ and $y$.

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$\begin{array}{l} \triangle P Q R\sim\triangle U V W\ \ \text { (Given) } \\\\ \displaystyle \frac{P Q}{U V}=\displaystyle \frac{Q R}{V W}=\displaystyle \frac{P R}{U W} \\\\ \displaystyle \frac{7}{x}=\displaystyle \frac{10}{y}=\displaystyle \frac{12}{9} \\\\ \displaystyle \frac{7}{x}=\displaystyle \frac{4}{3} \\\\ 4 x=21 \\\\ x=5.25 \\\\ \displaystyle \frac{10}{y}=\displaystyle \frac{4}{3} \\\\ 4 y=30 \\\\ y=7.5 \end{array}$

5.           The measures of two angles of $\triangle X Y Z$ are $82^{\circ}$ and $16^{\circ} .$ Find the measures of the angles of a triangle similar to $\triangle X Y Z$.

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The measures of the first two angles of $\triangle X Y Z$ are $82^{\circ}$ and $16^{\circ}$.

$\therefore$ The measure of the third angle of $\triangle X Y Z$ \[ \begin{array}{l} =180^{\circ}-\left(82^{\circ}+16^{\circ}\right)\\\\ =180^{\circ}-98^{\circ} \\\\ =82^{\circ} \end{array} \] $\therefore$ The measures of the angles of a triangle similar to $\triangle X Y Z$ are $82^{\circ}, 82^{\circ}$ and $16^{\circ}$ .