Question

Triangle ABC can be taken to triangle A′B′C′ using rigid motions and a dilation. Select all the equations that are true.

Answer 1

After a rigid motion and dilation, triangles ABC and A'B'C' are similar, then their corresponding sides are proportional.

So, we have the general equation:

$\frac{BA}{B^{\prime}A^{\prime}}=\frac{BC}{B^{\prime}C^{\prime}}=\frac{AC}{A^{\prime}C^{\prime}}$

So, let's evaluate each statement.

(a)

$\frac{A^{\prime}C^{\prime}}{B^{\prime}A^{\prime}}=\frac{AC}{BA}$

To evaluate this statement, let's choose the first and the third part of the general equation:

$\frac{BA}{B^{\prime}A^{\prime}}=\frac{AC}{A^{\prime}C^{\prime}}$

Multiplying both sides by A'C' /BA:

$\frac{BA}{B^{\prime}A^{\prime}}*\frac{A^{\prime}C^{\prime}}{BA}=\frac{AC}{A^{\prime}C^{\prime}}*\frac{A^{\prime}C^{\prime}}{BA}$

And solving the question:

$\frac{A^{\prime}C^{\prime}}{B^{\prime}A^{\prime}}=\frac{AC}{BA}$

So, the statement is true.

(b)

$\frac{B^{\prime}C^{\prime}}{B^{\prime}A^{\prime}}=\frac{BA}{BC}$

To evaluate this statement, let's choose the first and second terms of the general equation:

$\begin{gathered} \begin{equation*} \frac{BA}{B^{\prime}A^{\prime}}=\frac{BC}{B^{\prime}C^{\prime}} \end{equation*} \\ \end{gathered}$

Let's multiply both sides by B'C'/BA and solve the fractions.

$\begin{gathered} \frac{BA}{B^{\prime}A^{\prime}}*\frac{B^{\prime}C^{\prime}}{BA}=\frac{BC}{B^{\prime}C^{\prime}}*\frac{B^{\prime}C^{\prime}}{BA} \\ \frac{B^{\prime}C^{\prime}}{B^{\prime}A^{\prime}}=\frac{BC}{BA} \end{gathered}$

Since the expressions are different, the statement is false.

(c)

$\frac{AC}{A^{\prime}C^{\prime}}=\frac{B^{\prime}A}{BA}^$

To evaluate the statement, let's choose the first and the third part of the generic equation.

$\begin{gathered} \frac{BA}{B^{\prime}A^{\prime}}=\frac{AC}{A^{\prime}C^{\prime}} \\ which\text{ }is\text{ the }same\text{ }as: \\ \frac{AC}{A^{\prime}C^{\prime}}=\frac{BA}{B^{\prime}A^{\prime}} \end{gathered}$

Since the second part is not equal to the statement, the statement is false.

(d)

$\frac{CA}{C^{\prime}A^{\prime}}=\frac{CB}{C^{\prime}B^{\prime}}$

To evaluate this statement, let's choose the second and the third part of the equation.

$\begin{gathered} \begin{equation*} \frac{BC}{B^{\prime}C^{\prime}}=\frac{AC}{A^{\prime}C^{\prime}} \end{equation*} \\ which\text{ }is\text{ }the\text{ }same\text{ }as \\ \frac{AC}{A^{\prime}C^{\prime}}=\frac{BC}{B^{\prime}C^{\prime}} \\ wh\imaginaryI ch\text{ }is\text{ }the\text{ }same\text{ }as \\ \frac{CA}{C^{\prime}A}=\frac{CB}{C^{\prime}B^{\prime}} \end{gathered}$

So, the statement is true.

(e)

$\frac{A^{\prime}B^{\prime}}{AB}=\frac{C^{\prime}B^{\prime}}{C^{\prime}B}$

Let's evaluate the first and the second statement.

$\begin{gathered} \begin{equation*} \frac{BA}{B^{\prime}A^{\prime}}=\frac{BC}{B^{\prime}C^{\prime}} \end{equation*} \\ which\text{ }is\text{ }the\text{ }same\text{ }as \\ \frac{AB}{A^{\prime}B^{\prime}}=\frac{CB}{C^{\prime}B^{\prime}} \end{gathered}$

If a/b = c/d, then b/a = d/c. So,

$\frac{A^{\prime}B^{\prime}}{AB}=\frac{C^{\prime}B^{\prime}}{CB}$

The statement is true.

In summary,

The statements that are true are:

$\frac{A^{\prime}C^{\prime}}{B^{\prime}A^{\prime}}=\frac{AC}{BA}$$\frac{CA}{C^{\prime}A}=\frac{CB}{C^{\prime}B^{\prime}}$$\frac{A^{\prime}B^{\prime}}{AB}=\frac{C^{\prime}B^{\prime}}{CB}$