1. Given a point (−5, 4) on a geometric figure, what is the new point after rotating the figure 90 degrees counterclockwise?
2. Given point A(4,7) on the triangle, what are the new coordinates of the point after the triangle is rotated 180 degrees counterclockwise?
3. Triangle MNO is translated up 5 units and left 2 units. Point N is located at (4,-2). What are the coordinates of N′?
4. Triangle XYZ is translated down 4 units and to the left 8 units. The length of side XY is 10 units. What is the length of side X′Y′?
5. A line segment has endpoints (2,−3) and (1, −4). What are the new endpoints after rotating the segment 90° clockwise?
1. To rotate a point 90 degrees counterclockwise, we switch the coordinates and negate the new x-coordinate. So the new point after rotating (-5, 4) 90 degrees counterclockwise is (4, 5).
2. To rotate a point 180 degrees counterclockwise, we negate both coordinates. So the new coordinates of point A(4,7) after the triangle is rotated 180 degrees counterclockwise would be (-4, -7).
3. To translate a point up 5 units and left 2 units, we subtract 2 from the x-coordinate and add 5 to the y-coordinate. So the coordinates of N' would be (4 - 2, -2 + 5) = (2, 3).
4. Translating a triangle down 4 units and to the left 8 units does not change the length of its sides. So the length of side XY, which is 10 units, would remain the same as the length of side X'Y'.
5. To rotate a line segment 90 degrees clockwise, we switch the coordinates, negate the new y-coordinate, and switch the signs. So the new endpoints of the line segment with endpoints (2, -3) and (1, -4) after rotating 90 degrees clockwise would be (-(-4), 1) and (-(-3), 2), which simplifies to (4, 1) and (3, 2).