A polygon is shown on the graph:
A polygon is shown on the coordinate plane. Vertices are located at 3 comma 5, 1 comma 3, 1 comma 1, 2 comma 0, 4 comma 0, 5 comma 1, and 5 comma 3.
What effect will a translation 3 units down and 2 units left have on the polygon? Be sure to address how it could impact the angles, side lengths, and congruency between the original pre-image and the image. (5 points)
A translation of 3 units down and 2 units left will shift all the vertices of the polygon downward by 3 units and to the left by 2 units.
- Angle: The angles of the polygon will remain unchanged. Translation does not affect the angles of a shape, so the measures of the angles in the pre-image will be the same as the measures of the corresponding angles in the image.
- Side Lengths: The side lengths of the polygon will remain unchanged. Translation does not affect the lengths of the sides, so the lengths of the sides in the pre-image will be the same as the lengths of the corresponding sides in the image. The distance between any two points on a side of the polygon will be the same before and after the translation.
- Congruency: The pre-image and the image will be congruent. A translation is a rigid transformation that preserves both the angles and the lengths of the sides, so the polygon and its image will be congruent.
In summary, the translation will simply move the entire polygon downward by 3 units and to the left by 2 units, while keeping the angles and side lengths unchanged, resulting in a congruent polygon.
To find the effect of a translation on the polygon, you need to apply the same translation vector to each of the vertices. In this case, the translation is 3 units down and 2 units left.
To do this, subtract 3 from each y-coordinate and subtract 2 from each x-coordinate of the original vertices:
Original vertices:
A(3, 5)
B(1, 3)
C(1, 1)
D(2, 0)
E(4, 0)
F(5, 1)
G(5, 3)
New vertices after translation:
A'(1, 2)
B'(-1, 0)
C'(-1, -2)
D'(0, -3)
E'(2, -3)
F'(3, -2)
G'(3, 0)
Now, let's analyze the effects of the translation on the polygon:
1. Side lengths: The distances between the original vertices and their corresponding translated vertices will remain the same. So, the side lengths of the polygon will not be affected by the translation.
2. Angles: The translation does not change the shape or orientation of the polygon, so the angles between the sides will remain the same.
3. Congruency: The translation preserves the size and shape of the polygon, meaning the original polygon and its translated image are congruent. The original polygon and the translated polygon will have the same side lengths and angles.
In summary, after a translation 3 units down and 2 units left, the polygon will be shifted downward and to the left while maintaining its size, shape, side lengths, and angles.
To determine the effect of a translation 3 units down and 2 units left, we need to apply these transformations to each vertex of the polygon.
Original vertices:
A: (3, 5)
B: (1, 3)
C: (1, 1)
D: (2, 0)
E: (4, 0)
F: (5, 1)
G: (5, 3)
Applying the translation to each vertex:
A': (3 - 2, 5 - 3) = (1, 2)
B': (1 - 2, 3 - 3) = (-1, 0)
C': (1 - 2, 1 - 3) = (-1, -2)
D': (2 - 2, 0 - 3) = (0, -3)
E': (4 - 2, 0 - 3) = (2, -3)
F': (5 - 2, 1 - 3) = (3, -2)
G': (5 - 2, 3 - 3) = (3, 0)
The effect of the translation is that each vertex of the original polygon has been shifted 2 units to the left and 3 units down to create a new image.
Impact on angles:
The translation does not impact the angles of the polygon. The measure of each angle remains the same, as translation only affects the position of the vertices.
Impact on side lengths:
The translation does not affect the lengths of the sides of the polygon. The distance between each pair of vertices remains the same.
Congruency between pre-image and image:
The translation preserves congruency. The pre-image and the image are congruent polygons because the corresponding sides have the same lengths and the corresponding angles have the same measures.
In conclusion, the effect of the translation 3 units down and 2 units left on the given polygon is that each vertex is shifted to a new position while maintaining the same angles, side lengths, and congruency with the original pre-image.
A polygon is shown on the graph:
A polygon is shown on a coordinate plane. The vertices are A at negative 6 comma 5, B at negative 6 comma 2, C at negative 2 comma 2, and D at negative 2 comma 6.
If the polygon is translated 4 units down and 5 units right, what will the coordinates of the new image be? Use prime notation in expressing the new coordinates. (5 points)
To determine the coordinates of the new image after translating the polygon 4 units down and 5 units right, we need to add 5 to each x-coordinate and subtract 4 from each y-coordinate of the original vertices:
Original vertices:
A: (-6, 5)
B: (-6, 2)
C: (-2, 2)
D: (-2, 6)
New vertices after translation:
A': (-6 + 5, 5 - 4) = (-1, 1)
B': (-6 + 5, 2 - 4) = (-1, -2)
C': (-2 + 5, 2 - 4) = (3, -2)
D': (-2 + 5, 6 - 4) = (3, 2)
The coordinates of the new image are:
A': (-1, 1)
B': (-1, -2)
C': (3, -2)
D': (3, 2)
Therefore, the new image after the translation will have the vertices A' at (-1, 1), B' at (-1, -2), C' at (3, -2), and D' at (3, 2).