Polygon ABCD slides 4 units left and 3 units up on the coordinate plane. If the original measure of angle A was 75 degrees, what is the measure of angle A′? (4 points)
75 degrees
100 degrees
105 degrees
180 degrees
To determine the measure of angle A' after the polygon is translated 4 units left and 3 units up on the coordinate plane, you need to understand that translations do not affect the measures of angles. So, the measure of angle A' will still be 75 degrees.
Therefore, the correct answer is 75 degrees.
To find the measure of angle A', we need to determine the effect of the translation on the angle.
When a polygon is translated on a coordinate plane, the shape and size of the polygon remain the same, but its position changes.
In this case, the polygon ABCD was moved 4 units left and 3 units up. Therefore, the measure of angle A' will remain the same as the original measure of angle A, which is 75 degrees.
So, the correct answer is:
75 degrees
To find the measure of angle A' after the polygon is slid, we first need to determine the new coordinates of point A.
If the polygon slides 4 units to the left, the x-coordinate of point A changes from x to x-4. Similarly, if the polygon slides 3 units up, the y-coordinate of point A changes from y to y+3.
Now, we need to use basic trigonometry to find the measure of angle A. We can use the tangent function: tan(A) = (change in y) / (change in x).
In this case, the change in x is -4 (the polygon slides to the left), and the change in y is 3 (the polygon slides up).
tan(A) = (3) / (-4)
Now we take the arctan of both sides to find angle A:
A = arctan(3 / -4)
Using a calculator, we find: A ≈ -36.87 degrees
Since angle A' is just the same as angle A, the measure of angle A' is also approximately -36.87 degrees.
Therefore, the correct answer is not provided in the options.
The line segments are translated 2 units to the left to form A′B′ and C′D′. Which statement describes A′B′ and C′D′? (4 points)
Line segments A′B′ and C′D′ do not intersect and are closer together than AB and CD.
Line segments A′B′ and C′D′ intersect at (−2, 0) and are two times farther apart than AB and CD.
Line segments A′B′ and C′D′ intersect at (0, −2) and are two times closer together than AB and CD.
Line segments A′B′ and C′D′ do not intersect and are the same distance apart as AB and CD.
To find the new coordinates of A', B', C', and D' after the translation, we need to subtract 2 units from the x-coordinates of points A, B, C, and D.
Let's assume that the original coordinates of points A, B, C, and D are (x, y).
After the translation, the new coordinates of points A', B', C', and D' would be (x - 2, y).
Now, let's analyze the given answer choices:
1) Line segments A'B' and C'D' do not intersect and are closer together than AB and CD.
Based on the information given, we cannot determine whether A'B' and C'D' are closer together than AB and CD. This statement cannot be concluded.
2) Line segments A'B' and C'D' intersect at (-2, 0) and are two times farther apart than AB and CD.
Since A' and B' have their x-coordinates shifted by 2 units to the left (x - 2), and C' and D' also have their x-coordinates shifted by 2 units to the left (x - 2), the coordinates of the point of intersection of A'B' and C'D' would be (-2, 0). However, there is no information given about the distances between AB and CD, and A'B' and C'D'. So, we cannot conclude whether A'B' and C'D' are two times farther apart than AB and CD.
3) Line segments A'B' and C'D' intersect at (0, -2) and are two times closer together than AB and CD.
Again, there is no information provided to determine whether A'B' and C'D' are two times closer together than AB and CD. This statement cannot be concluded.
4) Line segments A'B' and C'D' do not intersect and are the same distance apart as AB and CD.
Since A', B', C', and D' have their x-coordinates shifted by 2 units to the left (x - 2) but their y-coordinates remain the same (y), the line segments A'B' and C'D' will be the same distance apart as AB and CD. Therefore, this statement is correct.
Hence, the correct answer is:
Line segments A′B′ and C′D′ do not intersect and are the same distance apart as AB and CD.
Which translation will change figure ABC to figure A′B′C′? (4 points)
A coordinate plane is shown. Figure ABC is composed of coordinates A, negative 4 comma negative 1, B, negative 1 comma negative 1, C, and negative 2 comma negative 3. Figure A prime, B prime, and C prime is composed of coordinates A prime, 0 comma 2, B prime, 3 comma 2, C prime, 2 comma 0.
3 units right and 3 units up
4 units right and 3 units up
4 units right and 4 units up
4 units left and 3 units down
To determine the translation that changes figure ABC to figure A'B'C', we need to find the difference in coordinates between the corresponding points.
The original coordinates for ABC are:
A(-4,-1)
B(-1,-1)
C(-2,-3)
The new coordinates for A'B'C' are:
A'(0,2)
B'(3,2)
C'(2,0)
To find the translation, we need to determine the change in x-coordinates and the change in y-coordinates between the original and new coordinates.
For point A:
Change in x-coordinate = 0 - (-4) = 4
Change in y-coordinate = 2 - (-1) = 3
For point B:
Change in x-coordinate = 3 - (-1) = 4
Change in y-coordinate = 2 - (-1) = 3
For point C:
Change in x-coordinate = 2 - (-2) = 4
Change in y-coordinate = 0 - (-3) = 3
Based on these calculations, we can see that the translation that changes figure ABC to figure A'B'C' is moving 4 units right and 3 units up.
Therefore, the correct answer is:
4 units right and 3 units up.
The figure shows Triangle ABC and some of its transformed images on a coordinate grid:
A coordinate plane is shown. Triangle ABC Is formed by the points 2 comma 5, 2 comma 3 and 4 comma 3. Triangle 2 is formed by the points negative 3 comma 5, negative 3 comma 3, and negative1 comma 3. Triangle 1 is formed by the points negative 4 comma 5, negative 4 comma 3, and negative 6 comma 3. Triangle 3 is formed by the points negative 6 comma 2, negative 4 comma 2, and negative 4 comma 0. Triangle 4 is formed by the points 2 comma negative 1, 4 comma negative 1 and 4 comma negative 3.
Which of the four triangles was formed by a translation of Triangle ABC? (4 points)
1
2
3
4
To determine which triangle was formed by a translation of Triangle ABC, we need to analyze the coordinates of the vertices of each triangle.
Triangle ABC:
A (2, 5)
B (2, 3)
C (4, 3)
Triangle 1:
A' (-4, 5)
B' (-4, 3)
C' (-6, 3)
Triangle 2:
A' (-3, 5)
B' (-3, 3)
C' (-1, 3)
Triangle 3:
A' (-6, 2)
B' (-4, 2)
C' (-4, 0)
Triangle 4:
A' (2, -1)
B' (4, -1)
C' (4, -3)
For a translation to occur, the coordinates of each point in the transformed triangle must have a consistent change in the x and y values compared to the original triangle.
Looking at the coordinates of Triangle 2, we can see that the x-coordinates have all shifted 5 units to the left (-3 - 2 = -5, -3 - 2 = -5, -1 - 4 = -5) and the y-coordinates have remained the same. This consistent change in the x-coordinates is indicative of a translation.
Therefore, Triangle 2 was formed by a translation of Triangle ABC.
Hence, the correct answer is triangle 2.