Transformation Combinations Practice Quiz

A translation moves every point by the same distance in a fixed direction, preserving the shape and size of the figure. There is no change in orientation or scale.

Reflection creates a mirror-image of a figure about a designated line called the line of reflection. It reverses the orientation while preserving distances.

A rotation turns a figure around a fixed point by a given angle, keeping the distance from the center the same. The overall shape and size remain intact during the rotation.

A dilation changes the size of a figure by a certain scale factor while maintaining its shape. The proportions of the figure remain constant, though its size increases or decreases.

The provided matrix is the standard representation of a rotation in the plane about the origin by an angle θ. It maintains distances and angles, hence preserving the figure's congruence.

Reflecting over y = x swaps the coordinates, and reflecting the result across the y-axis negates the new x-coordinate. This composition sends (x, y) to (-y, x), which is equivalent to a 90° counterclockwise rotation.

The dilation enlarges the figure uniformly by a factor of 2, and subsequently the translation shifts every point by the same vector. The order in this case results in an enlarged figure that is repositioned.

A glide reflection is a composite transformation combining a reflection with a translation along the direction of the reflecting line. It is the only isometry among these that results in no fixed point.

Rotation is an isometry that preserves both the shape and size of a figure. Although the figure's position and orientation change, the distances and angles within the figure remain the same.

When a figure is reflected over two parallel lines, the net effect is a translation. The distance of this translation is twice the distance between the two parallel lines.

A dilation with a scale factor less than 1 compresses the figure, reducing all distances proportionally. The original shape is maintained though the figure becomes smaller in size.

Both translation and reflection are isometries, and as such they preserve distances and angles within the figure. The composite transformation guarantees that the figure remains congruent to its original shape.

Rotating a point 180° about the origin sends it to the point diametrically opposite, effectively negating both coordinates. This is a direct consequence of the properties of rotation in the plane.

While the rotation and subsequent dilation preserve angles and the overall shape (similarity), the dilation alters the side lengths of the figure. As a result, the figure is no longer congruent to its original form.

Isometric transformations preserve the distances between points, ensuring that shapes remain congruent. Although other attributes like orientation and position may change, the intrinsic dimensions of the figure remain constant.

The rotation by 60° followed by a reflection over the line y = √3x produces a composite matrix that simplifies to [1/2, √3/2; √3/2, -1/2]. This matrix corresponds to a reflection about a line making a 30° angle with the positive x-axis, which is equivalent to reflecting across the line y = (1/√3)x.

The dilation first reduces (4, 3) to (2, 1.5). A 90° clockwise rotation then maps (2, 1.5) to (1.5, -2) by swapping coordinates and negating the new y-value, yielding the final image.

First performing a dilation enlarges the figure, and following it with a reflection produces a mirror image of this enlarged figure. The process changes the scale as well as the orientation, resulting in a non-isometric transformation overall.

A 270° counterclockwise rotation (equivalent to a 90° clockwise rotation) transforms (2, -1) to (-1, -2). The subsequent translation by (-3, 4) shifts it to (-4, 2), which is the final image of the point.

While isometries preserve shape, angles, and distances, a dilation changes the scale of the figure. As a result, the size and side lengths are not preserved, even though the overall shape remains similar.