Chapter 6.6, Problem 77E

Solution

To explain: The reason why the triangles formed by $0$ , $1$ , and $z_1$ , and by $0$ , $z_2$ , and $z_1{z}_2$ shown in the figure are similar triangles.

The triangles formed by $0$ , $1$ , and $z_1$ , and by $0$ , $z_2$ , and $z_1{z}_2$ shown in the figure are similar triangles because triangles follows the SAS similarity.

Given Information:

The triangles formed by $0$ , $1$ , and $z_1$ , and by $0$ , $z_2$ , and $z_1{z}_2$ shown in the figure are similar triangles.

Calculation:

  $z_1$ has a modulus of $r_1$ and an angle of ${\theta }_1$ with respect to the $x$ -axis. $z_2$ has a modulus of $r_2$ and an angle of ${\theta }_2$ with respect to the $x$ -axis. $z_1{z}_2$ has a modulus of $r_1{r}_2$ and an angle of ${\theta }_1+{\theta }_2$ with respect to the $x$ -axis. So, we can label triangles as shown below.

  

The larger triangle has an angle at the origin which has a measure of $\left({\theta }_1+{\theta }_2\right)-{\theta }_2={\theta }_1$ . The smaller triangle has an angle at the origin which has a measure of ${\theta }_1$ .

The ratio of the corresponding sides that include ${\theta }_1$ between the two triangles are:

  $\frac{r_1{r}_2}{r_1}=r_2\left(\text{for the longer sides}\right)$

And,

  $\frac{r_2}{1}=r_2\left(\text{for the medium sides}\right)$

Since two corresponding side lengths are proportional and their included angles are congruent, then the two triangles are similar by SAS similarity.

Therefore, the triangles formed by $0$ , $1$ , and $z_1$ , and by $0$ , $z_2$ , and $z_1{z}_2$ shown in the figure are similar triangles because triangles follows the SAS similarity.