Question

rectangle abcd is transformed to rectangle abcd. find the sequence of transformations. grid image showing rectangles abcd (purple) and abcd (red) with coordinate axes options: translated 5 units left and 8 units down; translated 8 units left and 5 units down; translated 5 units down and 8 units right; translated 8 units left and 5 units up

Explanation:

Step1: Identify coordinates of A and A'

Point A: Let's find the coordinates of point A in the original rectangle (purple). From the grid, A is at (2, y). Wait, looking at the y - axis, the original rectangle is above the x - axis. Let's check the x - coordinate: A is at x = 2? Wait no, looking at the grid, the original rectangle ABCD: A is at (2, let's see the y - grid. Wait, the red rectangle A'B'C'D' is below the x - axis. Let's take point A (original) and A' (transformed).

Original A: Let's look at the grid. The original rectangle (purple) has A at (2, 5)? Wait, no, let's check the x - coordinate: A is at x = 2? Wait, the x - axis has numbers from - 8 to 8. The original rectangle's A: x - coordinate is 2? Wait, no, looking at the grid, the original rectangle ABCD: A is at (2, 5)? Wait, the red rectangle A'B'C'D' has A' at (-6, -1)? Wait, no, let's do it properly.

Let's take point A: Original A (purple) is at (2, 5)? Wait, no, the x - coordinate of A: looking at the grid, the vertical line for x = 2 passes through D. Wait, maybe I made a mistake. Let's take point B. Original B is at (6, 5) (since it's on x = 6, y - coordinate: let's see the grid lines. The original rectangle is above the x - axis, so y - coordinate is positive. The red rectangle is below the x - axis, y - coordinate negative.

Point A (original): Let's find the coordinates. From the grid, the original rectangle (purple) has A at (2, 5) (x = 2, y = 5). Point A' (red) is at (-6, -1)? Wait, no, let's check the x - coordinate of A': A' is at x = - 6, y = - 1? Wait, no, the red rectangle A'B'C'D' has A' at (-6, -1)? Wait, maybe better to take the horizontal and vertical shifts.

Horizontal shift (left/right): The original x - coordinate of A: let's see, A is at x = 2 (since D is at x = 2). A' is at x = - 6. So the change in x: - 6 - 2 = - 8. So that's a shift of 8 units to the left (since x decreases by 8).

Vertical shift (up/down): Original y - coordinate of A: let's see, A is above the x - axis. Let's say A is at y = 5 (since the rectangle has height, let's check the red rectangle. A' is below the x - axis. Let's say original A is at y = 5, A' is at y = 0? No, wait, the red rectangle is below the x - axis. Wait, maybe A is at (2, 5) and A' is at (-6, 0)? No, that's not right. Wait, let's take point A:

Original A: (2, 5) (x = 2, y = 5)
A': (-6, 0)? No, that's not. Wait, maybe I messed up the y - coordinate. Let's look at the grid lines. The original rectangle (purple) is from y = 1 to y = 5? No, the red rectangle (A'B'C'D') is from y = - 5 to y = - 1? Wait, let's count the units.

From A (original) to A' (transformed):

Horizontal: Original x of A: let's see, the original rectangle's A is at x = 2 (since D is at x = 2). A' is at x = - 6. So 2 - (-6) = 8? Wait, no, to go from x = 2 to x = - 6, we subtract 8 (2 - 8 = - 6), so that's a shift of 8 units left.

Vertical: Original y of A: let's say A is at y = 5 (above x - axis). A' is at y = 0? No, the red rectangle is below the x - axis. Wait, A' is at y = - 1? Wait, no, the red rectangle is below the x - axis, so y - coordinate is negative. Let's count the vertical units. From y = 5 (original A) to y = 0? No, wait, the original rectangle's height: from y = 1 to y = 5? No, let's take point A and A':

Original A: (2, 5)
A': (-6, 0)? No, that's not. Wait, maybe I should take point B. Original B is at (6, 5), A' is at (-6, -1)? No, this is confusing. Wait, let's use the formula for translation: if a point (x, y) is translated h units left/right (h < 0 for left, h > 0 for right) and k units up/down (k > 0 for up, k < 0 for down), the new point is (x + h, y + k).

Let's take point A:

Original A: Let's find its coordinates. Looking at the grid, the original rectangle (purple) has A at (2, 5) (x = 2, y = 5). The transformed A' is at (-6, 0)? No, wait, the red rectangle A'B'C'D' has A' at (-6, -1)? Wait, no, the red rectangle is below the x - axis. Let's check the y - coordinate: the original rectangle is above the x - axis (y positive), the red is below (y negative). So the vertical shift is down.

Let's take point A (original) at (2, 5) and A' (transformed) at (-6, 0)? No, that's not. Wait, maybe I made a mistake in the y - coordinate. Let's look at the grid lines. The original rectangle (purple) is from y = 1 to y = 5 (so height 4). The red rectangle (A'B'C'D') is from y = - 5 to y = - 1 (height 4). So the vertical shift: from y = 5 to y = - 1: 5 - (-1) = 6? No, that's not. Wait, maybe the original A is at (2, 4) and A' is at (-6, -1). Then the horizontal shift: - 6 - 2 = - 8 (8 units left), vertical shift: - 1 - 4 = - 5 (5 units down). Ah! That makes sense. So original A (2, 4), A' (-6, -1). So x - shift: - 6 - 2 = - 8 (8 units left), y - shift: - 1 - 4 = - 5 (5 units down). So the translation is 8 units left and 5 units down.

Step2: Verify with another point

Let's take point B (original) at (6, 4). Translating 8 units left: 6 - 8 = - 2. Translating 5 units down: 4 - 5 = - 1. So B' should be at (-2, -1). Looking at the grid, B' is at (-2, -1), which matches. So the translation is 8 units left and 5 units down.

Answer:

Translated 8 units left and 5 units down