By a translation of axes, points 𝐒(x1, y1) and 𝐓(x2, y2) are given new coordinates (x1^', y1^')

step 1 First problem we're going to let P equal ST be a point in the xy plane. We'll let p prime equal

By a translation of axes, points 𝐒(x1, y1) and 𝐓(x2, y2) are...

By a translation of axes, points $\mathbf{S}\left(x_1, y_1\right)$ and $\mathbf{T}\left(x_2, y_2\right)$ are given new coordinates $\left(x_1^{\prime}, y_1^{\prime}\right)$ and $\left(x_2^{\prime}, y_2^{\prime}\right)$, respectively, in accordance with Equations (2) on page 171. Verify algebraically that $$ \sqrt{\left(x_2^{\prime}-x_1^{\prime}\right)^2+\left(y_2^{\prime}-y_1^{\prime}\right)^2}=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2} $$ and interpret this fact geometrically.

By a translation of axes, points $\mathbf{S}\left(x_1, y_1\right)$ and $\mathbf{T}\left(x_2, y_2\right)$ are given new coordinates $\left(x_1^{\prime}, y_1^{\prime}\right)$ and $\left(x_2^{\prime}, y_2^{\prime}\right)$, respectively, in accordance with Equations (2) on page 171. Verify algebraically that

$$
\sqrt{\left(x_2^{\prime}-x_1^{\prime}\right)^2+\left(y_2^{\prime}-y_1^{\prime}\right)^2}=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}
$$

and interpret this fact geometrically.

Key Concepts

Euclidean Distance and the Distance Formula

The Euclidean distance between two points is computed using the distance formula, which involves the square root of the sum of the squared differences of the coordinates. This formula measures the straight-line or 'as the crow flies' distance between the points, and it is fundamental in describing spatial relationships in a Cartesian coordinate system.

Rigid Motion/Isometry

A rigid motion, or isometry, is a transformation that preserves distances and angles. Since translations do not distort the relative positions of points, they qualify as rigid motions. This concept is crucial in understanding that the geometric properties of figures remain unchanged under such transformations.

Translation of Axes

The translation of axes is a transformation where every point in the coordinate system is shifted by a fixed vector. This means that the original coordinates of a point are adjusted by adding constant values, resulting in new coordinates that are relative to a shifted origin.

Invariance of Distance Under Translation

Invariance under translation refers to the property that a translation of the coordinate system does not affect the distances between points. When the same constant vector is added to both points, the differences in their coordinates remain unchanged, ensuring that the computed Euclidean distance is the same before and after the translation.

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