Geometry Chapter 9 Test Part I

The translation (x,y) → (x + 7, y - 5) means that each point in the original triangle is shifted 7 units to the right and 5 units down to get the corresponding point in the image triangle. Using this translation, the coordinates of X(-5,2) become X’(2,-3), the coordinates of Y(0,-4) become Y’(7,-9), and the coordinates of Z(3,3) become Z’(10,-2). Therefore, the vertices of the image triangle are X’(2,-3), Y’(7,-9), and Z’(10,-2).

Figure B does not have rotational symmetry because it cannot be rotated by any angle and still look the same. The other figures, A, C, and D, have rotational symmetry because they can be rotated by certain angles (such as 90 degrees or 180 degrees) and still maintain their original appearance.

A 90º counterclockwise rotation about the origin means that each point on the plane is rotated 90º in the counterclockwise direction around the origin. To find the image of (1,-6) after this rotation, we can switch the x and y coordinates and negate the new x coordinate. Switching the coordinates gives us (-6,1), and negating the x coordinate gives us (6,1). Therefore, the image of (1,-6) for a 90º counterclockwise rotation about the origin is (6,1).

If the preimage is in Quadrant II and the image is in Quadrant III, it means that the reflection is happening across the x-axis. This is because the x-axis divides the coordinate plane into two halves - the positive y-axis and the negative y-axis. When a point is reflected across the x-axis, its y-coordinate changes sign while the x-coordinate remains the same. Therefore, the line of reflection in this case is the x-axis.

The moon takes about 27 1/3 days to complete one journey around Earth. Since the motion of the moon can be modeled by a rotation, a 90º angle of rotation would represent one-fourth of the moon's journey. Therefore, the moon completes a 90º angle of rotation in about 7 days (27 1/3 divided by 4).

The correct answer is (x,y) → (x - 12, y + 4). This rule describes a translation where the point is moved 4 units up and 12 units to the left. The negative sign in front of the 12 indicates that the translation is to the left, and the positive sign in front of the 4 indicates that the translation is up.

The reflection image of a point across a line can be found by finding the point of intersection between the line and the perpendicular line passing through the given point. In this case, the line y = -x has a slope of -1. The perpendicular line passing through (5, -3) will have a slope of 1. Using the point-slope form of a line, we can find the equation of the perpendicular line as y - (-3) = 1(x - 5), which simplifies to y = x - 8. Solving the system of equations y = -x and y = x - 8, we find the point of intersection to be (3, -5). Therefore, the reflection image of (5, -3) across the line y = -x is (3, -5).

The reflection image of a point across the y-axis involves changing the sign of the x-coordinate while keeping the y-coordinate the same. In this case, the given point (5,-3) is reflected across the y-axis, resulting in the point (-5,-3).

The motion of the moon around the Earth can be modeled as a circle rotation about a point. The point of rotation is the center of the Earth, and the moon rotates around this point in a circular path. This is because the same hemisphere of the moon always faces the Earth, indicating a rotation around a fixed point. The angle of rotation is determined by the time interval, which is approximately 27 1/3 days for one journey of the moon around the Earth.

A rectangle has exactly two lines of symmetry. A line of symmetry is a line that divides the figure into two equal halves that are mirror images of each other. In a rectangle, the two lines of symmetry are the vertical line passing through the center of the rectangle and the horizontal line passing through the center. This means that if you fold the rectangle along these lines, the two halves will perfectly overlap each other. Therefore, the correct answer is rectangle.

A regular hexagon has six equal sides and six equal angles. In order to rotate the hexagon onto itself, we need to find the smallest angle that brings the hexagon back to its original position. Since the hexagon has six equal angles, we can divide 360º (the total angle measure of a hexagon) by 6 to find the measure of each angle, which is 60º. Therefore, the smallest angle through which we can rotate a regular hexagon onto itself is 60º.

Rotational symmetry is not a symmetry for the tessellation because a tessellation is a pattern that repeats without any rotation. Rotational symmetry refers to the property of an object or shape being able to be rotated and still look the same. In a tessellation, the pattern remains the same when it is translated or reflected, but not when it is rotated. Therefore, rotational symmetry is not applicable to tessellations.

The translation (x,y) → (x + 3,y + 2) shifts every point 3 units to the right and 2 units up. To find the preimage, we need to reverse this translation. Starting from the given point (5,-9), we need to move 3 units to the left and 2 units down to find the original point. Therefore, the preimage is (5 - 3, -9 - 2), which simplifies to (2, -11).

The translation moves the point P(11,-5) 12 units to the left and 6 units down, resulting in the point (-1, -11). The reflection in x = 0 means that the point is reflected over the y-axis, resulting in the point (1, -11). Therefore, the correct answer is (1, -11).

When a regular hexagon is rotated counterclockwise by 240º about its center, the image of any point on the hexagon will be located at a position that is two-thirds of the way around the hexagon in the counterclockwise direction. In this case, the image of point X will be located at point G.

When we reflect a figure in the y-axis, the x-coordinates of all the points are negated while the y-coordinates remain the same. When we reflect the resulting figure in the x-axis, the y-coordinates of all the points are negated while the x-coordinates remain the same. This sequence of transformations effectively flips the figure over both the x-axis and the y-axis.

On the other hand, a rotation of 90º followed by a reflection in the x-axis would involve rotating the figure 90º counterclockwise and then flipping it over the x-axis. These two transformations do not result in the same figure as the given sequence of reflections in the y-axis and x-axis. Therefore, the correct answer is a rotation of 90º followed by a reflection in the x-axis.

A parallelogram has rotational symmetry but not reflectional symmetry because it can be rotated by 180 degrees and still appear the same, but it cannot be reflected to create a mirror image of itself. This is because a parallelogram has two pairs of parallel sides, which allows it to be rotated without changing its appearance. However, when a parallelogram is reflected, the sides are no longer parallel, resulting in a different shape.

A plane can be tessellated using a regular octagon together with a square. The regular octagon and square can be arranged in a pattern where each octagon shares a side with a square, creating a repeating pattern that covers the entire plane without any gaps or overlaps. This is a valid tessellation because the octagon and square have the same side lengths and angles, allowing them to fit together perfectly.

A figure tessellates a plane if it can be repeated infinitely without any gaps or overlaps to cover the entire plane. Looking at the figure B, it has irregular shapes and angles which will not fit together perfectly to cover the plane without gaps or overlaps. Therefore, figure B will not tessellate a plane.

The reflection image of a point across a line can be found by reflecting the point over the line. In this case, the line of reflection is y = -6. To find the reflection image of (a,b) across this line, we need to keep the x-coordinate the same and change the sign of the y-coordinate. Therefore, the correct answer is (a, -12 - b).