Question

select the correct answer from each drop - down menu. rectangle abcd is dilated by a scale factor of \\(\frac{1}{4}\\) to form abcd. point t is the center of dilation and lies on line segment ab as shown. in abcd, line segment ab has a slope of \\(\boxed{}\\) and a length of \\(\boxed{}\\). reset next

Explanation:

Step1: Determine Slope of AB

First, find coordinates of A and B. From graph, A is (4, 4) and B is (6, 4)? Wait, no, looking at the grid: Wait, the y-axis and x-axis: Wait, the graph has y-axis (vertical) and x-axis (horizontal). Wait, point A: let's check the grid. Wait, the rectangle ABCD: D is (2, 2)? Wait, no, maybe I misread. Wait, the y-axis is labeled with 0, 2, 4, 6 (upwards) and x-axis 0, 2, 4, 6, 8 (rightwards). Wait, point D: (2, 2)? No, point D is at (2, 2)? Wait, point A: (4, 4)? Point B: (6, 4)? Wait, no, the rectangle: A and B are horizontal, D and C are horizontal. Wait, AB is a horizontal line? Because A and B have the same y-coordinate. So slope of AB is 0 (since slope = (y2 - y1)/(x2 - x1), if y2 = y1, slope is 0). Dilation preserves the slope, so A'B' will also have slope 0.

Step2: Determine Length of AB

First, find length of AB. Let's find coordinates of A and B. Let's assume A is (4, 4) and B is (6, 4)? Wait, no, looking at the grid: the distance between A and B: if A is (4, 4) and B is (6, 4), then the length is |6 - 4| = 2? Wait, no, maybe A is (4, 4) and B is (6, 4)? Wait, the grid squares: each square is 1 unit? So from x=4 to x=6, that's 2 units? Wait, no, maybe A is (4, 4) and B is (6, 4), so length AB is 2? Wait, no, maybe I made a mistake. Wait, the rectangle: D is (2, 2), A is (4, 4)? No, D is at (2, 2), A is at (4, 4)? Wait, no, the coordinates: let's check the y-axis (vertical) and x-axis (horizontal). The point A is at (4, 4) (x=4, y=4)? Wait, no, the y-axis is vertical, so the first coordinate is x, second is y? Wait, no, standard coordinate system: (x, y), where x is horizontal (right), y is vertical (up). So in the graph, the y-axis is labeled with 0, 2, 4, 6 (upwards) and x-axis 0, 2, 4, 6, 8 (rightwards). So point D: (2, 2)? No, point D is at (2, 2)? Wait, point A: (4, 4)? Point B: (6, 4)? Point C: (6, 6)? Point D: (4, 6)? Wait, no, that would be a square. Wait, maybe A is (4, 4), B is (6, 4), C is (6, 6), D is (4, 6). Then AB is from (4, 4) to (6, 4), so length is 6 - 4 = 2? Wait, no, 6 - 4 is 2? Wait, no, 6 - 4 is 2 units? Wait, no, if each grid square is 1 unit, then from x=4 to x=6 is 2 units. Then dilation by scale factor 1/4: length of A'B' is (1/4)length of AB. If AB is 4 units? Wait, maybe I messed up the coordinates. Wait, let's re-express: suppose A is (4, 4) and B is (8, 4)? No, the x-axis goes up to 8. Wait, the rectangle: D is (2, 2), A is (4, 4), B is (8, 4), C is (8, 2)? No, that would be a rectangle. Wait, no, the original rectangle ABCD: D is (2, 2), A is (4, 4), B is (6, 4), C is (6, 2)? No, that's a square. Wait, maybe AB is from (4, 4) to (8, 4), so length 4? Then dilation by 1/4: length A'B' is 4(1/4) = 1? Wait, no, let's check again. Wait, the problem says "line segment AB" – in the original rectangle, AB is horizontal (since A and B have same y-coordinate), so slope is 0. Dilation preserves slope, so A'B' also has slope 0. Now, length of AB: let's find coordinates of A and B. Let's assume A is (4, 4) and B is (8, 4) (so length 4). Then dilation by 1/4: length A'B' = 4(1/4) = 1. Wait, but maybe AB is length 4? Let's confirm: if A is (4, 4) and B is (8, 4), then the distance is 8 - 4 = 4. Then dilation factor 1/4: 4(1/4) = 1. So slope of A'B' is 0 (same as AB, since dilation preserves parallelism and slope for horizontal lines), and length is 1.

Answer:

Slope: 0, Length: 1