Question

1. describe how to distinguish between an isometry and a transformation that is not an isometry.
2. write a function to describe each translation. each solid figure is a pre - image, and each dashed figure is an image.

a.
b.

Explanation:

Step1: Define isometry

An isometry is a transformation that preserves distances and angles. So, check if distances between points and angle - measures remain the same before and after transformation. If they do, it's an isometry; if not, it's not an isometry.

Step2: For part 2a

Let the translation vector from point $P$ to point $Q$ be $\vec{v}=(x_Q - x_P,y_Q - y_P)$. If a point $(x,y)$ in the pre - image is translated, the function for the translation is $T(x,y)=(x+(x_Q - x_P),y+(y_Q - y_P))$.

Step3: For part 2b

Let the translation vector from point $Q$ to point $P$ be $\vec{u}=(x_P - x_Q,y_P - y_Q)$. The function for the translation of a point $(x,y)$ in the pre - image is $T(x,y)=(x+(x_P - x_Q),y+(y_P - y_Q))$.

Answer:

1. Check if distances between points and angle - measures are preserved. If they are, it's an isometry; if not, it's not.

2.
a. $T(x,y)=(x+(x_Q - x_P),y+(y_Q - y_P))$
b. $T(x,y)=(x+(x_P - x_Q),y+(y_P - y_Q))$