Short Answer
Note: For questions 19 and 20, your teacher will grade your response to ensure you receive proper credit for your answer.
Point A(4, 2) is translated according to the rule (x, y) right arrow (x + 1, y – 5) and then reflected across the y-axis.
a) In which quadrant of the coordinate plane is point A located?
b) What are the coordinates of translated point A’? In which quadrant of the coordinate plane is point A’ located?
c) What are the coordinates of reflected point A’’? In which quadrant of the coordinate plane is point A’’ located?
If the coordinates of an ordered pair have opposite signs, then one coordinate must be positive and the other coordinate must be negative. This can only happen in two quadrants: the second quadrant and the fourth quadrant.
In the second quadrant, the x-coordinate (which is negative) is decreasing from left to right, and the y-coordinate (which is positive) is increasing from bottom to top.
In the fourth quadrant, the x-coordinate (which is positive) is increasing from left to right, and the y-coordinate (which is negative) is decreasing from bottom to top.
Therefore, if the coordinates of an ordered pair have opposite signs, the pair must lie either in the second quadrant or the fourth quadrant.
a) Point A is located in the first quadrant of the coordinate plane.
b) The coordinates of translated point A' are (5, -3) and it is located in the fourth quadrant of the coordinate plane.
c) The coordinates of reflected point A'' are (-5, -3) and it is located in the second quadrant of the coordinate plane.
Can the figure below tessellate a plane? Explain your answer.
A shape is formed by cutting out a smaller semicircle from a larger semicircle. The remaining closed figure resembling an arch is shown.
No, the figure cannot tessellate a plane. A tessellation is a pattern of repeating shapes that completely cover a plane without any gaps or overlaps. The given figure has curved edges and cannot be rotated or reflected to fit together with identical copies of itself without gaps or overlaps. Therefore, it cannot tessellate a plane.
bot Note: For questions 21–22, your teacher will grade your response to ensure you receive proper credit for your answer.
Look at the shaded triangle in the drawing below. Describe in detail the transformations needed to create the rest of the square design. You can use a combination of translations, reflections, and rotations of the shaded triangle (Figure I) to create the square design.
To create the rest of the square design, we can perform the following transformations on the shaded triangle (Figure I):
1. Reflect the shaded triangle across the vertical axis (the y-axis) to get the second triangle (Figure II).
2. Rotate the second triangle 90 degrees clockwise about its center to get the third triangle (Figure III).
3. Reflect the third triangle across the horizontal axis (the x-axis) to get the fourth triangle (Figure IV).
4. Translate the fourth triangle to the right and up by the same distance as the height of the original shaded triangle to get the fifth triangle (Figure V).
5. Reflect the fifth triangle across the vertical axis (the y-axis) to get the sixth triangle (Figure VI).
6. Rotate the sixth triangle 90 degrees clockwise about its center to get the seventh triangle (Figure VII).
7. Reflect the seventh triangle across the horizontal axis (the x-axis) to get the final triangle (Figure VIII), which completes the square design.
Note: Make sure to describe each transformation clearly, including the axis, angle, and distance of rotation and the direction and distance of translation.
bot he coordinates of an ordered pair have opposite signs. In which quadrant(s) must the ordered pair lie? Explain.
Grayson is designing a rectangular athletic field. On his scale drawing, the vertices of the rectangle are (4, 5), (9, 10), and (9, 5). What are the coordinates of the fourth vertex?
A. (4, 9)
B. (4, 10)
C. (5, 10)
D. (5, 9)
The fourth vertex of the rectangle will have the same y-coordinate as the first vertex because the opposite sides of a rectangle are parallel. The y-coordinate of the first vertex is 5, so we know the y-coordinate of the fourth vertex is also 5.
To find the x-coordinate of the fourth vertex, we can use the fact that the opposite sides of a rectangle are congruent. The length of the horizontal side of the rectangle is the difference between the x-coordinates of the first and second vertices, which is 9 - 4 = 5. Therefore, the length of the horizontal side of the rectangle is also 5.
Since the fourth vertex has the same y-coordinate as the first vertex, its x-coordinate must be 5 units to the left of the x-coordinate of the second vertex, which is 9. Therefore, the x-coordinate of the fourth vertex is 9 - 5 = 4.
So the coordinates of the fourth vertex are (4, 5), which corresponds to answer choice A.
A. (–3, 33)
B. (8, –98)
C. (–9, 105)
D. (3, –39)