Question

the rectangle bcde is a dilation of the rectangle bcde. what is the scale factor of the dilation? simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Find length of BCDE's side

Take vertical side of BCDE (e.g., from B to E). B is at ( - 4, 8), E at ( - 4, - 8). Length: \( |8 - (-8)| = 16 \)? Wait, no, maybe horizontal or vertical segment. Wait, B'C'D'E' is blue, BCDE red. Let's take vertical side: B' is at (-3,6)? Wait, grid: each square is 1 unit. B is at (-4,8), B' at (-3,6)? Wait no, looking at y - coordinates: B is at y = 8, B' at y = 6? Wait, no, the blue rectangle B'C'D'E': B' is at (-3,6)? Wait, no, let's check coordinates. Let's take the vertical side of BCDE: from B ( - 4, 8) to E ( - 4, - 8)? No, that's too long. Wait, maybe the vertical side from B to E: B is ( - 4, 8), E is ( - 4, - 8)? No, the red rectangle: B is (-4,8), C is (4,8), D is (4,-8), E is (-4,-8)? No, that can't be. Wait, no, the red rectangle: B is (-4,8), C is (4,8), D is (4,-8), E is (-4,-8)? No, the blue rectangle: B' is (-3,6), C' is (3,6), D' is (3,-6), E' is (-3,-6)? Wait, no, looking at the graph: B is at (-4,8), B' at (-3,6)? No, maybe the vertical segment from B to E: B is ( - 4, 8), E is ( - 4, - 8)? No, the length of BCDE's vertical side: from y = 8 to y = - 8, that's 16? No, the blue rectangle: from y = 6 to y = - 6, length 12? Wait, no, let's take a side. Let's take the vertical side of BCDE: B is at ( - 4, 8), E is at ( - 4, - 8)? No, that's not right. Wait, the red rectangle: B is (-4,8), C is (4,8), D is (4,-8), E is (-4,-8). The blue rectangle: B' is (-3,6), C' is (3,6), D' is (3,-6), E' is (-3,-6). Wait, no, maybe the horizontal side? No, dilation scale factor is ratio of image length to original length. Let's take the vertical side: original (BCDE) vertical length: from y = 8 to y = - 8? No, that's 16. Image (B'C'D'E') vertical length: from y = 6 to y = - 6? No, that's 12. Wait, no, maybe I messed up. Wait, B is at (-4,8), B' at (-3,6)? No, looking at the graph: B is at x = - 4, y = 8; B' is at x = - 3, y = 6? No, the grid lines: x from -10 to 10, y from -10 to 10. Each grid square is 1 unit. So B is at ( - 4, 8), B' is at ( - 3, 6)? No, the blue dot B' is at x = - 3, y = 6? Wait, no, the x - coordinate: B is at x = - 4, B' at x = - 3? No, the distance between B and B' horizontally? No, dilation is about a center, probably the origin? Wait, scale factor is (length of image) / (length of original). Let's take the vertical side of BCDE: from B ( - 4, 8) to E ( - 4, - 8). Wait, no, E is at ( - 4, - 8)? Then length is 8 - ( - 8) = 16. The vertical side of B'C'D'E': from B' ( - 3, 6) to E' ( - 3, - 6). Length is 6 - ( - 6) = 12. 12 / 16 = 3/4? No, that's not. Wait, maybe I took the wrong side. Wait, B is at ( - 4, 8), C is at (4, 8). So horizontal side BC: length \( |4 - (-4)| = 8 \). B' is at ( - 3, 6), C' at (3, 6). Horizontal side B'C': length \( |3 - (-3)| = 6 \). Then scale factor: 6 / 8 = 3/4? Wait, no, 6/8 is 3/4? Wait, 6 divided by 8 is 3/4. Wait, or vertical side: B to E: from y = 8 to y = - 8, length 16. B' to E': from y = 6 to y = - 6, length 12. 12/16 = 3/4. Yes, that makes sense. So scale factor is 6/8 or 12/16, which simplifies to 3/4? Wait, no, wait: B is at (-4,8), B' at (-3,6)? No, wait the x - coordinate of B is - 4, B' is - 3? No, looking at the graph, the blue rectangle: B' is at x = - 3? No, the grid: each square is 1 unit. Let's check the coordinates again. B (red) is at ( - 4, 8), B' (blue) is at ( - 3, 6)? No, the x - coordinate of B is - 4, B' is - 3? No, the distance between B and B' in x: 1 unit, y: 2 units. No, dilation scale factor is (image length) / (original length). Let's take the vertical segment from B ( - 4, 8) to E ( - 4, - 8): length is 16 (from y = - 8 to y = 8). The vertical segment from B' ( - 3, 6) to E' ( - 3, - 6): length is 12 (from y = - 6 to y = 6). 12 / 16 = 3/4. Alternatively, horizontal segment: B ( - 4, 8) to C (4, 8): length 8. B' ( - 3, 6) to C' (3, 6): length 6. 6 / 8 = 3/4. Yes, so scale factor is 3/4? Wait, no, wait: 6 divided by 8 is 3/4. So the scale factor is 3/4? Wait, but let's confirm. Dilation scale factor is (length of image) / (length of pre - image). So pre - image BCDE, image B'C'D'E'. So length of B'C' over BC: 6/8 = 3/4. So scale factor is 3/4.

Step2: Confirm the scale factor

We took two sides: horizontal (BC and B'C') and vertical (BE and B'E'). Both give 6/8 = 3/4 and 12/16 = 3/4. So the scale factor is 3/4.

Answer:

\(\frac{3}{4}\)