Essay
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.
A square is drawn on a coordinate plane with the center of the square on the origin. Two diagonals are drawn through the origin forming four congruent triangles on the inside of the square. The triangle that overlaps the positive y-axis is shaded.
To create the rest of the square design, we can use a combination of translations, reflections, and rotations of the shaded triangle. The following steps can be taken:
1. Reflect the shaded triangle over the y-axis.
2. Rotate the reflected triangle 90 degrees counterclockwise around the origin.
3. Translate the triangle 2 units to the right.
4. Reflect the translated triangle over the x-axis.
5. Rotate the reflected triangle 180 degrees around the origin.
6. Translate the triangle 2 units up.
7. Reflect the translated triangle over the y-axis.
8. Rotate the reflected triangle 90 degrees clockwise around the origin.
9. Translate the triangle 2 units to the left.
10. Reflect the translated triangle over the x-axis.
11. Rotate the reflected triangle 180 degrees around the origin.
12. Translate the triangle 2 units down.
13. Reflect the final triangle over the y-axis to obtain the complete square design.
In summary, the transformations needed to create the rest of the square design are: reflection over y-axis, 90 degree counterclockwise rotation, translate 2 units to the right, reflection over x-axis, 180 degree rotation, translate 2 units up, reflection over y-axis, 90 degree clockwise rotation, translate 2 units to the left, reflection over x-axis, 180 degree rotation, translate 2 units down, and final reflection over y-axis.
The coordinates of an ordered pair have opposite signs. In which quadrant(s) must the ordered pair lie? Explain.
If an ordered pair has opposite signs, that means one coordinate is positive and the other is negative. The only two quadrants where this is possible are the second quadrant and the fourth quadrant.
In the second quadrant, the x-coordinate (which is negative) is decreasing as you move to the left and the y-coordinate (which is positive) is increasing as you move up. So any ordered pair in the second quadrant will have opposite signs.
In the fourth quadrant, the x-coordinate (which is positive) is increasing as you move to the right and the y-coordinate (which is negative) is decreasing as you move down. So any ordered pair in the fourth quadrant will also have opposite signs.
Therefore, if an ordered pair has opposite signs, it must lie in either the second quadrant or the fourth quadrant.
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
Hallie is trying to win the grand prize on a game show. Should she try her luck by spinning a wheel with 6 equal sections labeled from 1 to 6 and hope she gets a 5, or should she roll two number cubes and hope she gets the same number on both cubes? Explain
The probability of spinning a 5 on the wheel is 1/6 because there is only one section labeled 5 out of 6 equal sections.
The probability of rolling the same number on two number cubes is 1/6 as well, but calculated differently. There are six possible outcomes for the first roll, and each of those six outcomes can be paired with six possible outcomes for the second roll. So there are 6 x 6 = 36 equally likely outcomes when rolling two number cubes. Of those 36 outcomes, there are six outcomes where both cubes show a 1, six outcomes where both cubes show a 2, and so on up to six outcomes where both cubes show a 6. That means there are a total of 6 + 6 + 6 + 6 + 6 + 6 = 36 outcomes where the two cubes show the same number.
Comparing the probabilities, we can see that the probability of rolling the same number on two number cubes is greater than the probability of spinning a 5 on the wheel. Therefore, Hallie should roll two number cubes and hope to get the same number on both cubes if she wants the better chance of winning the grand prize.
To create the rest of the square design, we need to perform a series of transformations on the shaded triangle (Figure I). We can use a combination of translations, reflections, and rotations.
1. Translation:
First, we need to translate the shaded triangle horizontally and vertically to create copies of it in the other three quadrants of the square. We can do this by moving the shaded triangle to the right along the x-axis and to the top along the y-axis. Copy the shaded triangle and place it in the second quadrant (top-left), third quadrant (bottom-left), and fourth quadrant (bottom-right) of the square.
2. Reflection:
Next, we need to reflect the shaded triangle across the x-axis and y-axis. To reflect across the x-axis, imagine flipping the shaded triangle upside down, so the positive y-axis becomes the negative y-axis. Copy the reflected triangle and place it in the third quadrant (bottom-left) and the fourth quadrant (bottom-right) of the square. To reflect across the y-axis, imagine flipping the shaded triangle horizontally, so the positive x-axis becomes the negative x-axis. Copy the reflected triangle and place it in the second quadrant (top-left) and the fourth quadrant (bottom-right) of the square.
3. Rotation:
Finally, we need to rotate the shaded triangle 180 degrees around the origin to complete the square design. Copy the shaded triangle and rotate it 180 degrees, placing it in the third quadrant (bottom-left) of the square.
By performing these transformations, you will create the rest of the square design, using translations, reflections, and rotations of the shaded triangle (Figure I).